Nicola Scafetta: Jupiter’s orbital eccentricity may drive ~60yr and millennial climate cycles.

Posted: September 24, 2020 by tallbloke in solar system dynamics
Figure 3. The 60‐year eccentricity function (blue) of Jupiter (see Figure 2) against: (a) the HadCRUT global surface temperature record (Morice et al., 2012) detrended of its quadratic polynomial fit f(t) ¼ a(t − 1850)2 + b (cf. Scafetta, 2010, 2016) (correlation coefficient r^2 = 0:5, p < 0.01); (b) the 5‐year running average of the Indian summer monsoon rainfall from 1813 to 1998 (Agnihotri & Dutta, 2003) (correlation coefficient r^2 = 0:5, p < 0.01)


 Plain Language Summary 

The physical origin of the modulation of the cloud system and of many of the Earth’s climate oscillations from the decadal to the millennial timescales is still unclear, despite its importance in climate science. One of the most prominent oscillations has a period of about 60 years and is found in a number of geophysical records such as temperature reconstructions, aurora sights, Indian rainfalls, ocean climatic records, and in many others. These oscillations might emerge from the internal variability of the climate system, but increasing evidence also points toward a solar or astronomical origin.

Herein we speculate whether the oscillations of the orbits of the planetary system could modulate the interplanetary dust flux falling on the Earth, then modifying the cloud coverage. We find that the orbital eccentricity of Jupiter presents a strong 60‐year oscillation that is well correlated with several climatic records and with the 60‐year oscillation found in long meteorite fall records since the 7th century. Since meteorite falls are the most macroscopic aspect of infalling space dust, we conclude that the interplanetary dust should modulate the formation of the clouds and, thus, drive climate changes.

Scafetta, N., Milani, F., & Bianchini, A. (2020).
A 60‐year cycle in the Meteorite fall frequency suggests a possible
interplanetary dust forcing of the Earth’s climate driven by planetary
oscillations.

Geophysical Research Letters, 47, e2020GL089954.
https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2020GL089954

Personal study copy here

Comments
  1. pochas94 says:

    Kudos to Scafetta for pushing the science forward undaunted. I nominate cosmic rays for additional consideration. They dry out the radiating zone, lower the emitting height, and cool both the emitting altitude and the surface.

  2. tallbloke says:

    Yes, Scafetta leaves the Svensmark theory on the table in the paper, and offers his Jupiter-directed interplanetary dust as an additional mechanism.

  3. oldbrew says:

    Re. auroras:

    A shared frequency set between the historical mid-latitude aurora records and the global surface temperature (2012)
    Nicola Scafetta

    The existence of a natural 60-year modulation of the global surface temperature induced by astronomical mechanisms, by alone, would imply that at least 60-70% of the warming observed since 1970 has been naturally induced. Moreover, the climate may stay approximately stable during the next decades because the 60-year cycle has entered in its cooling phase.

    Paper here.

  4. tallbloke says:

    “Thousands of tons of cosmic dust are estimated to reach the Earth’s surface every year,[3] with most grains having a mass between 10−16 kg (0.1 pg) and 10−4 kg (100 mg).[3] ”

    https://en.wikipedia.org/wiki/Cosmic_dust

  5. Recent generations of the world’s scientists seem to have absorbed an exaggerated dependence upon models founded upon suggestive graphs like the above. The first point to be made, always, is how good is the r-square (correllation coefficient, to we older scientists), which tells one directly how much of the variation in the supposed-dependent variable is due to variations in the supposed causal variable. The above graph fails — in this older-generation scientist’s view — by not giving that r-square value (“r 1/4 0:5, p<0.01" is non-informative, even meaningless to the directly-informative — and simple(!) — r-square value). The observed variation — the noise — in the temperature record is suspicious, and suggests the true r-square value of the comparison is small, and hence untrustworthy.

    Second, the temperature record is "detrended of its quadratic polynomial fit", which obviously (looking at the graph) means the global warming signal has been removed. So, in Scarfatta's model, there is no global warming, and his presumed cause of global warming, cosmic dust seeding of clouds, is disallowed.

    The supposed 60-year temperature cycle has been famously explained as multidecadal ocean temperature oscillations, but that is with the global warming signal included. And with that inclusion, is the only real motivation for even believing in the global temperature records of HADCRUT and others over the last century. But there is plenty of evidence of incompetent and/or criminal handling of the temperature records, by politically compromised and debased climate scientists. So in reality, the multidecadal ocean oscillations theory is itself suspect, and with it that 60-year supposed cycle in the global mean surface temperature.

    Sorry, but Scarfatta is playing with pretty stones on the beach, while ignorant of the great ocean of truth nearby. It is all a great waste of peoples' time and energy…par for the course today. In reality, there is no global warming, and thus no 60 year cycle in that global warming, and no control of the global mean temperature either by ocean oscillations or by cosmic dust variations.

    And that of course is what I, starting with my Venus/Earth temperatures comparison, brought out 10 years ago: There is no valid global climate science, and no competent climate scientists.

  6. oldbrew says:

    Just came across an article about Nobel prize winner CTR Wilson…includes a dust-related experiment.

    The cloud chamber and CTR Wilson’s legacy to atmospheric science

    Introduction and early life

    2011 is the centenary year of the short
    paper (Wilson, 1911) first describing the
    cloud chamber, the device for visualising
    high-energy charged particles which earned
    the Scottish physicist Charles Thomas Rees
    (‘CTR’) Wilson the 1927 Nobel Prize for physics.
    His many achievements in atmospheric
    science, some of which have current relevance,
    are briefly reviewed here.

    CTR Wilson’s lifetime of scientific research work
    was principally in atmospheric electricity at
    the Cavendish Laboratory, Cambridge; he
    was Reader in Electrical Meteorology from
    1918 and Jacksonian Professor from 1925 to
    1935. However, he is immortalised in physics
    for his invention of the cloud chamber,
    because of its great significance as an early
    visualisation tool for particles such as cosmic rays
    (Galison, 1997).
    . . .
    In summary, CTR Wilson’s visualisation
    techniques for particle physics concerned
    microscopic cloud processes, whereas his
    synthesis of atmospheric electricity unravelled
    invisible atmospheric properties on a
    global scale. Half a century after his death,
    it is a tribute to his painstaking reasoning
    and wonderful experimental ingenuity that
    both his principal scientific achievements
    still influence physics education and atmospheric
    electricity research.

    Click to access Harrison2011_CTRWilson_Weather.pdf

    Only a few pages but worth a read IMO.

  7. tallbloke says:

    If Harry had taken the time to read the linked full paper, he’d have found that …..

    Actually, I can’t be bothered to spoon feed info to people who know it all without reading it.

    Suffice to say the 60 year cycle with detrended T data is supplemented by the millennial cycle which accounts for the centennial ups and downs apparent in longer term T reconstructions..

  8. Paul Vaughan says:

    The following commentary has NOTHING to do with planets, orbits, or temperatures.
    It’s just the conventional paradigm.

    r^2 alone does NOT determine what statisticians call “statistical significance”.
    A low r^2 can be highly statistically significant.
    A high r^2 can fail to be statistically significant.

  9. Paul Vaughan says:

    Irretrievable: 2 comments vanished.

    Political commentary on the “60 year linear component” of climate variables — motivated not by climate so far as I can tell but maybe fear that western financial sanctions won’t have a valid military guarantor by 2030 or 2036 — ignores the more general context of nonlinear slip cycles.

    Alert readers may have noticed.

    “The number filter” readily accepts political rants (even if they are wild). Johnson’s, Macron’s, & Biden’s people will be happy to note: At worst IT’s a very strong participation turn-off and IT’s at least a cause of major “behavior modification” (ware IT may be inverted totalitarianism).

    I have some “mysterious” (only “thanks” to the west turn 2030 “backup energy” plan, but otherwise acutely precise) calculations I would share, but for sure they won’t pass the filter.

    Harry would no doubt be left with unanswered questions about the block of numbers and IT’s creative decode.

  10. tallbloke says:

    Paul, was this a wordpress issue? No comments in spam to be found, which is where they end up when the ‘vanish’ on submission.

  11. Paul Vaughan says:

    One was on the recent XR thread and one on the 2400 year thread. I must have an out-of-date e-mail address for you.

  12. ren says:

    Cosmic dust can be held in the upper stratosphere for a long time by jet currents. It should be concentrated in the equatorial belt, which may limit the heating of the tropical ocean.

  13. oldbrew says:

    Scafetta’s paper says:
    The periodogram shows that the eccentricity function is characterized by a prominent 922‐year oscillation. Jupiter’s eccentricity also shows another prominent 60‐year oscillation (apparently made of two cycles at about 57.1 and 60.9 years), which is slightly modulated by the 20‐year conjunction cycle with Saturn.

    The 922-year oscillation looks a lot like the lunar precession period at 1/7th of 6441 tropical years, or 104 apsidal cycles, as described at the Talkshop here:

    For every 104 apsidal cycles, all numbers except SM slip by -1 from being multiples of 104, i.e. a precession. So after 7*104 LAC all the other totals except SM are ‘reduced’ by 7 each.

    https://tallbloke.wordpress.com/2017/10/15/lunar-precession-update/
    [SM = synodic month]

    6441/7 = 920 + 1/7 tropical years.

    To explain it another way, the formula of 55 full moon cycles = 7 apsidal cycles = 62 tropical years (55+7=62) is almost true. But the more accurate formula requires multiplying by 104:
    55*104, -7 FMC = 7*104 apsidal = 62*104, -7 TY

    Hence 1/7th of that is one lunar precession period, or whatever you want to call it.
    7 apsidal cycles = 766 synodic months.
    – – –
    Re the 60.9 years, a suggestion would be that this is really the time of 360 degrees of rotation of the Jupiter-Saturn conjunction (~61.05 tropical years), another precession period.

  14. tallbloke says:

    OB. Good spot. Considering the relative masses of Jupiter and our moon, it seems likely that it’s Jupiter’s 922yr eccentricity cycle which has shaped our moon’s precession cycle. The interesting question is whether it actually our moon which is channelling the distribution of cosmic dust in the upper atmosphere, as well as Jupiter channeling the particulate masses in our direction from interplanetary space.

  15. oldbrew says:

    Eccentricity and apsides…

    ECCENTRICITY EXCITATION AND APSIDAL RESONANCE CAPTURE IN THE
    PLANETARY SYSTEM u ANDROMEDAE [2002]

    This paper lays the groundwork for understanding the
    origin of the large eccentricities and the apsidal alignment
    exhibited by the orbits of planets C and D in u And. Our
    main result is that the eccentricity of planet C and the
    locking of orbital apsides are both consequences of the slow
    growth of the eccentricity of planet D. The latter eccentricity,
    in turn, was driven by an external agent—plausibly a
    primordial circumstellar disk lying exterior to the orbit of
    planet D—that acted over timescales exceeding 104 yr. We
    play our scenario out and explain the mechanics of apsidal
    resonance capture in [section] 2.

    Paper: ‘https://iopscience.iop.org/article/10.1086/341617/pdf’

  16. oldbrew says:

    From another Scafetta paper:
    Multiscale Analysis of the Instantaneous Eccentricity Oscillations of the Planets of the Solar System from 13 000 BC to 17 000 AD

    The eccentricity function of the orbit of Jupiter presents large oscillations with periods of about 60 and 900- 960 years, mostly due to the interaction with Saturn. These oscillations also correspond to oscillations found in several geophysical records. The eccentricity functions of Uranus and Neptune are characterized by a large 4300-year oscillation. The eccentricity function of Pluto is characterized by a large nearly 20000-year modulation.

    https://link.springer.com/article/10.1134/S1063773719110094

    The U-N eccentricity of 4300 years is very similar to:
    (U-N * N) / (U-N – N) = 4270.11~ years

    No. of N is 1 greater than no. of U-N in this period, so the planetary orientation relative to the Sun repeats, being a multiple of 360 degrees of movement.

  17. tallbloke says:

    6441/3 ~= 4270/2

  18. oldbrew says:

    4270 * 13/12 = ~4626

  19. Paul Vaughan says:

    4270 is the aliquot sum for B (baby monster 4370). Way too much is piling up to share — supremely rich vein.

  20. Paul Vaughan says:

    =
    φ(n)
    Euler Totient
    1584
    λ(n)
    Carmichael Lambda
    396

    There are 1,584 positive integers (less than 4,370) that are coprime with 4,370.
    =
    https://metanumbers.com/4370

    You’ll find those numbers buried in 104-yielding levels (once the moderator frees the most recent filter victim on 2400).

    Fits in what they call simple sporadic 5-group or Mathieu group M11 with order 7920.

    It’s a monstrous insight avalanche. The filter is a big problem. It hates round-off bars and math.

    Weather by dark deception or dark ignorance (dark either way) the so-called “experts” savagely misdirected US.

    Beyond B there are 2 ways to construct 1/(U-N) with M. When I start to post the fifth-roots stuff the filter will always be full.

    More mundane but should be noted: Scafetta should be looking at 66 not 61. You can’t differentiate between them with 2 waves, but it’s clear when you go back further.

    “EU’V-E got a monde stir in Eur. Paris Sol.
    The walls are closing in again: O[r]well.”
    — Queens of the Stone Age

  21. Paul Vaughan says:

    UN Seek Cure IT Count Sol.integrity

    General Ramanujan points to sharp spikes in discrete-continuous relations.

    For example, consider level 54 “almost-integer” (math lingo — what they call it) spike on 5th-root scale:

    R(p) = ⌊(e^π*54^(1/5))^(1/p)⌉^p – e^π*54^(1/5)

    Pervasive round-off brackets of discrete-continuous relations needn’t freeze reader awareness in blank-stares of ignorance.

    Link to easy answers for an example. If you can round off a number, you’re in.

    R(3) = -71.0083263199169 = ⌊(e^π*54^(1/5))^(1/3)⌉^3 – e^π*54^(1/5) = 10^3 – e^π*54^(1/5)

    Note 2 embedded links in the expression.

    Bracket expressions conservatively for the calculator — otherwise the automation is prone to silently making incorrect organizational changes (downscale fractal of misled big tech, no. doubt).

    At the end of the first link change the example 3 to 5.
    Round off the result to 4. That gives new input 4^5 for the beginning of the second link.

    Repeat with 10 at the right end of the first link.
    The result rounds off too 2 so 2^10 goes.in to the far left of the second link. Find:

    R(5) = R(10) = -47.0083263199169

    59 = ( 71 + 47 ) / 2
    Recall M: 47*59*71.

    Check R(6) / 2 for a clue as to how we nearly won over U-N (wins even tie won).

    DCoy daze review:
    171.406964273337 = 50 / ( 1/(2*(13*11*7*5*3)^(1/4)) + 2/(2*(13*11*7*5*3)^(1/4)) + 1/(2*(7*5*3)^(1/4)) )
    47.0085558422417 = ( 2 / ( 1/(2*(13*11*7*5*3)^(1/4)) + 2/(2*(13*11*7*5*3)^(1/4)) + 1/(2*(7*5*3)^(1/4)) ) ) ^ 2

    TB’11 also remember from way back in the day: (φ√5)^4 = 25*φ^4.
    171.352549156242 = 25*1.61803398874989^4

    Monde Stir’11 each let US anyon?
    No. Eur. CRude type 0! symbols O[r]We’11 build a C[ENSO]Rship mess tory, in.deed.

    Binet and Lucas AImost redirect what curry US IT to 196883196560 = 323 as fall lows:
    25*√(φ^8+Φ^8) = 25*√47
    171.391365010026 = 25*(1.61803398874989^8+0.618033988749895^8)^(1/2)

    EUCRUS: Stand buy northern D-fence 4 clarification.

    Note 646 = 2 * 323

    Half before the US election — and 1/2 after. My way of proving I’m know partisan under threat of financial terror directed by well-off e/11 IT WHO’s winter CR(U-N)chess west turn common folks numb brrrs right down to street-level aware noose weather left or right.

    OB’11 sea fib’n’luc.in top line …sequel too:

    171.391380036748
    = 5^2*(ΦΦ/(1/34/2-1/76/2))^(1/2) = 5^2*(ΦΦ*152*68/(152-68))^(1/2)
    = 5^2*(ΦΦ/(1/646+1/152))^(1/2) = 5^2*(ΦΦ*646*152/(646+152))^(1/2)

    152: wise IT AImost X[R(p)]act weather we sea 2400 or knot?

    171.391365010026 = 5^2*(ROUND(4^2*(59)^(1/2),0)-ROUND(3^2*(71)^(1/2),0))^(1/2)
    = 5^2*(123-76)^(1/2) = 5^2*(47)^(1/2)

    Shh! OK?.in’ 47 Tops B

    Aliquot sum (sum of proper divisors) for 2*5*19*23 = 4370:
    4270 = 1+2+5+10+19+23+38+46+95+115+190+230+437+874+2185

    “Velvet Ears” 123 Flash mnemonIC Back.in Hazard Count: Tie Wan M$y$n D-Po11ace

    171.406545820013 = 5^2*(EXP(54^(1/5)*PI())-ROUND(EXP(54^(1/5)*PI())^(1/5),0)^5)^(1/2)
    164.791478352793 = 1 / ( 1 / 4270 + 1/25/(EXP(54^(1/5)*PI())-ROUND(EXP(54^(1/5)*PI())^(1/5),0)^5)^(1/2) )
    84.016966353458 = 1 / ( 1 / 4270 + 2/25/(EXP(54^(1/5)*PI())-ROUND(EXP(54^(1/5)*PI())^(1/5),0)^5)^(1/2) )
    55.6463431474333 = 1 / (2 / 4270 + 3/25/(EXP(54^(1/5)*PI())-ROUND(EXP(54^(1/5)*PI())^(1/5),0)^5)^(1/2) )

  22. Paul Vaughan says:

    Study with care: easy to derive system at IC ally (it’s only a few trivial steps).

    You’ll recognize not only 2^(1/J/2) & φ^4*2^(1+1/J/2) but exactly why they’re off and by exactly how much. IT’s Ramanujan’s world of AImost.in toujours.

    836.531021854751
    835.546575435631
    0.117820651543 = % “error” (not actu[s]ally un error)

    61.0241156298752
    61.0464822565173
    -0.036638682223 = % “error”

    “Why fall O? 2 “higher” grrOun-D? 11Ost as USware ayaM?” — C-elective Sol

  23. Paul Vaughan says:

    By part tie sun IC top knew trail’ski left M is tory…

    836.531021854751 = φ^4*2^(1+11.8626151546089/2)
    835.546575435627 = 1/(5/29.4474984673838-2/11.8626151546089)

    61.0241156298752 = 2^(11.8626151546089/2)
    61.0464822565173 = 1/(1/11.8626151546089-2/29.4474984673838)

    710003.731008934 = (836.531021854751)*(835.546575435627) / (836.531021854751 – 835.546575435627)
    418.019254422807 = (836.531021854751)*(835.546575435627) / (836.531021854751 + 835.546575435627)
    836.038508845615 = (836.531021854751)*(835.546575435627)/((836.531021854751+835.546575435627)/2)

    71.0003731008934 = 710003.731008934 / 10000
    209.009627211404 = 418.019254422807 / 2

    …distinct from 208.

  24. Paul Vaughan says:

    The Simplest Conventional View

    836.492470214859 = 2/(1/418-1/710000)
    835.508109310293 = 2/(1/418+1/710000)
    61.0213033256461 = Φ^4/(1/418-1/710000)
    11.8624821779712 = 2*LOG(Φ^4/(1/418-1/710000),2)
    29.4471611398395 = 5/((1/418+1/710000)/2+1/LOG(Φ^4/(1/418-1/710000),2))
    61.0458600788091 = 1/(1/2/5/LOG(Φ^4/(1/418-1/710000),2)-(1/418+1/710000)/5)
    19.8648166947302 = 1/(3/2/5/LOG(Φ^4/(1/418-1/710000),2)-(1/418+1/710000)/5/2)
    8.45605035913211 = 1/(7/2/5/LOG(Φ^4/(1/418-1/710000),2)+(1/418+1/710000)/5/2)

    -0.001125389948 = % error average across j, s, beat, & axial with stable bias — sign & magnitude both stable meaning correction’s simple

    Out of curiosity compare the biased 1/(J-S) with:
    19.8643454852672 = 4 / ( 1/(2*(13*11*7*5*3)^(1/4)) + 1/(2*(7*5*3)^(1/4)) )
    0.002372136869 = % error

    Remember that 836 is the smallest untouchable weird number (not making this shh!IT up — it’s number theory lingo).

  25. Paul Vaughan says:

    Hi! K.in.on Boris Bluff

    Too daze note U-N.Doors.cores perfect non.11.in.ear.IT.
    Hears the backs tory: dec.aids a go sum BRIteechairs D-sided school-D B e/11.in.ear.

    28 is perfect.
    proper divisors of 298: 1, 2, 149
    s(298) = 1 + 2 + 149 = 152

    25 + 298 = 323 ——————– / —————————–

    See.in’re:cure.sieve.a11yET?

    From where again did Ramanujan claim many of his sharpest insights came?

    Other perfect numbers include 2, 5, 52, 88, & 96.
    UNassum.in2400 review spill28IT /

    Does every 1 remember whale engine.in’s voice on the ducts of has heard? This ain’t no.ware born Gen.R’a11[y]each let US.

  26. Paul Vaughan says:

    Freedom is Not a Dr.ill

    Ballparks can B Bo ring but knot fin weighin’ best tune where EU can see the playbook writin’ on “the green monster” weather left and right:

    28 = (84)*(42) / (84 + 42) = (1^2+2^2+3^2+…+22^2+23^2+24^2)^(1/2) – 42 = 70 – 42

    58 = 73+(-15)^(1)
    298 = 73+(-15)^(2)

    R(5) = 224.991673680083
    836 / 44 ~= -R(6) / R(2) = s(77)

    Artist IC types can’t be reached with just logic.

    146 = 298 – s(298) = 298 – 152
    s(48) = s(146) = 152 / 2 = 76

    Jovian Giant C-elective:
    Make sure you understand 298 as the (red hot) “others hide” of 25 (chili peppers).
    IT AIn’t 11.in.ear: climb IT SAM pulls cont.in.EU.US exponential with discrete s.wit.ch.

    General Summary

    M B Lee.ch.in jsun: a11 won and the same.

  27. Paul Vaughan says:

    “Spam” in the C[ENSO]Rship

    Can mod phi niche fish file tour?

    104 ~= ⌊(e^√s(298+25)π)^(1/2)⌉^2 – e^√s(298+25
    104 ~= ⌊(e^√s(196883-196560)π)^(1/2)⌉^2 – e^√s(196883-196560

    mod 13

  28. Paul Vaughan says:

    Note the Tall spike above.

    s(323) = 37
    -103.999977946281 = ⌊(e^√37π)^(1/2)⌉^2 – e^√37π

    Scenic aside:
    s(67) = 1
    s(2*67) = 70 ——– Leech
    s(3*67) = 71 ——– M
    s(4*67) = 208
    s(5*67) = 73 ——– lowest prime congruent to 1 mod 24
    s(8*67) = 11*44 = 22^2

    The order of presentation is cryptic, but the ingredients are well-served.

    Review.

  29. Paul Vaughan says:

    Luck key mods comparatively fish pike threw D-baitless lines of clean discretion.

    s(3*43) = 47
    -743.999775171279 = ⌊(e^√43π)^(1/3)⌉^3 – e^√43π
    s(3*67) = 71
    -743.999816894531 = ⌊(e^√67π)^(1/3)⌉^3 – e^√67π

    mod 8

    59 = s((59 mod 24)*s(3*43)) = s(s(20)*s(3*(67-24))/2) = (47+71)/2

  30. Paul Vaughan says:

    IT’s possible no. thing get$11earned bye the right UN tell after USelection.

    s(3*19) = 23 = 71 mod 24 = 47 mod 24
    = 196883 – ( ⌊(e^√19*(2π))^(1/2)⌉^2 – ⌊(e^√19*(2π))⌉ ) / 2
    -743.777680155239 = ⌊(e^√19π)^(1/3)⌉^3 – e^√19π

    What’s left tune out ice?

    s(6) = 6 = 3#

    “There are only four all-harshad numbers: 1, 2, 4, and 6 (The number 12 is a harshad number in all bases except octal).”

  31. Paul Vaughan says:

    Link to List of Aliquot Sums

    11.in.ear think kings ware IT just AIn’t .

    2432 / 836 = s(152/2) / s(76/2)
    84 = ( 836 + s(836) ) / s(34)

    s(30) = 42
    s(5#) = 42 where # indicates primorial
    5# = 5*3*2 product of all primes lower than or equal to 5

    18.6 = 744 / s(44)
    Suggestions-43 distilled moon lightin’ the table.

    Maybe only God Nos.: how many ways can Conway & Norton B written?
    298 = s(104/4)^2 + s(5#)
    = 146 + 2*s(146) = 146 + s(298) = 146 + 104 + s(47^2)

    Monster US aliquot sum phine PR O-ducts:

    φ = 2*cos(s(71^2)/s(59^2)/s(47^2)*8*π)
    2.61803398874989 = (2*cos(s(71^2)/s(59^2)/s(47^2)*s(2*5)*π))^2

    “Freedom I hold dear:
    The autumn moon lights my weigh
    — 11ed 22plan “R(amble)ln”

    Perfect construct$yen a head: list of aliquot sums s(n)

    s(28) = 28

    IT AIn’t just No. 1. WHO nos.?

    s(5*31) = 37 = s(196883-196560)
    s(7*29) = 37
    s(13*23) = 37
    s(17*19) = 37 = s(298+25)

  32. Paul Vaughan says:

    Vague O-Port Tune IT AIn’t

    Typo: “Other perfect numbers include 2, 5, 52, 88, & 96.” — correct shh!UNtouchABLE

    For more insight: Compare perfect with Ore.

    Once you memorize some of the key aliquot sums a whole (nonlinear) framework crystallizes. Remember what Conway and Norton said.

  33. Paul Vaughan says:

    Primorials (including 210) are no mystery at this stage, but there remains opportunity to clarify that 208 & 209 are both distinct and compatible. Above I outlined the most naive conventional model.

    On the 2400 thread I posted a very sharp calculation that no doubt puzzled pretty much anyone who looked at it dismissively — if only because they haven’t yet started to think carefully about discrete-continuous relations and aggregation criteria more generally.

    This insight eluded us for too long:

    208 ~ = slip(slip(24.067904774739,19.8650360864628),19.8650360864628)/2

    Once I saw that I understood the natural role of B & M in solar system stability at a whole new level.

    I used to hesitate — sometimes for months compounding delays from previous months — to post slip cycle calculations because the filter hates them. 2 comments never appeared. Technology failures have forced complete change of how, what — and when — to express.

    208
    209
    distinct
    and compatible

    Similarly we’re now past 5256/”836″/2. If you didn’t notice this yet: check this calculation with the 2 pairs of “836” above (pair from most naive conventional model and pair from Seidelmann (1992)).

    I defined what I mean by slip(x,y) on the 2400 thread. This convention along with chopping comments (especially ones with round-off bars) into fragments seems to keep the filter from blocking.

  34. Paul Vaughan says:

    Wikipedia’s “Kepler Trigon” overview isn’t based on Seidelmann (1992), which gives this slip series:
    1 / 19.8650360864628 = +1J-1S
    1 / 61.0464822565173 = +1J-2S
    1 / 835.546575435627 = -2J+5S
    1 / 2669.94916589798 = -29J+72S
    1 / 13660.3670170363 = +85J-211S
    1 / 117417.893491061 = -454J+1127S
    1 / 290290.064137486 = -4171J+10354S

    On the 2400 thread I applied the generalized Bollinger method to derive slip cycles including 66 & 132.

    Slip cycles are key (in long-run central limit) because circulatory structures and materials are aliasing exponentials. Our education system was designed to brainwash people into becoming dumb linear thinkers (puppets on straightforward strings). Fool me once shame on you fool me twice shame on me sort of thing.

    As some of you realize I no longer concern myself with “climate debate” but the context gives opportunity for astute readers to consider that hierarchies like the one listed at the beginning of this comment fall apart by the 2nd level with loose aggregation criteria. It isn’t necessarily just nonlinear drift on a curve; it’s potentially discrete switch-flipping.

    The comparative study of sets of “almost integer” (math lingo) fits alerts us to the existence of a higher organizing principle pulling threads towards a central limit. With this awakening we can realize several strands approaching (but not reaching) limits. A valid unifying principle accommodates such bundles — i.e. different combinations of pieces giving very-nearly the same thing.

  35. Paul Vaughan says:

    GA11actIC Green Monster Challenge 9801.in the C[ENSO]Rship

    “…and IT’s whisper-D that soon if we a11 ca11 the tune” — 11ed 22plan

    4th comm.in simplification & clarification suggests diagnostic comparison
    of Seidelmann’s (1992)
    2432 = 19 * 2^7
    with
    2436 = 29 * 84
    2436 = 58 * 42 ——— General√(Φ-φ)Ramanujan’s Guide to the 9*11*IX*XI = N.in.8.O.won

  36. Paul Vaughan says:

    Simplifying

    1/Φ = φ = ((1+5^(1/2))/2)

    11.8626176385713 = 1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+(1/104+1/298)*φ^2)
    29.4474891061275 = 1/(1/104+1/298)/φ^2

    19.8650473122013 = 20/(1/104+1/(1+(1/104+1/298)*φ^2/12))
    8.45614623658195 = 1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+2*(1/104+1/298)*φ^2)

    16.9122924731639 = 2/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+2*(1/104+1/298)*φ^2)

    104 = 1/((1/104+1/(1+(1/104+1/298)*φ^2/12))-1/(1+(1/104+1/298)*φ^2/12))

  37. Paul Vaughan says:

    Clarifying

    208 & 209 are thus distinct.

    61.0466285002156 = 1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20-(1/104+1/298)*φ^2)
    835.484250291355 = 1/((1/104+1/298)*φ^2-2*((1/104+1/(1+(1/104+1/298)*φ^2/12))/20-(1/104+1/298)*φ^2))

    61.0241681640784 = 2^(1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+(1/104+1/298)*φ^2)/2)
    836.531742004323 = φ^4*2^(1+1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+(1/104+1/298)*φ^2)/2)

    836.0076680293 = 2/(((1/104+1/298)*φ^2-2*((1/104+1/(1+(1/104+1/298)*φ^2/12))/20-(1/104+1/298)*φ^2))+1/φ^4/2^(1+1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+(1/104+1/298)*φ^2)/2))

    209.001917007325 = 1/(((1/104+1/298)*φ^2-2*((1/104+1/(1+(1/104+1/298)*φ^2/12))/20-(1/104+1/298)*φ^2))+1/φ^4/2^(1+1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+(1/104+1/298)*φ^2)/2))/2

    Familiar structures “almost” matching this frame afford rich review with developed hindsight.

  38. Paul Vaughan says:

    Level 7 Keys Concisely

    104 ~= slip(slip(24.067904774739,19.8650360864628),19.8650360864628)/4
    24.067904774739 = ⌊(e^√7π)^(1/p)⌉^p – e^√7π for p=2,3,4,6,12
    Twice the reciprocal of that for p = 1.

    This should help organize simple perspective because that (which keeps higher slips locked better) may look confusing.

    This approximation slips a little at higher-level, but being so simple (parsimony’s why I’m illustrating) it aids comparative study:
    24.067917505387 = 24+2*(1/104+1/298)*φ^2
    24th harmonic (divide period by 24) :
    1.00282989605779 = 1+(1/104+1/298)*φ^2/12
    0.993252365610063 = (104)*(1.00282989605779) / (104 + 1.00282989605779)
    20th subharmonic (multiply period by 20) ~= 1/(J-S)
    Level 7 has some really special properties. There are 2 other ways to estimate 24.0679 (outlined previously). Like 104 it’s a master key.

    For the set j, s, js beat, js axial, u, n, un beat, un axial arising from the last calculations above:
    0.000017200983 = average absolute %error

    The form extending j & s to u & n (based on perfect numbers 28 & 496) was outlined here.

    aside: flashback with hindsight
    5256.08395035091 = φ^4*2^(1+1/((1/104+1/(1+(1/104+1/298)*φ^2/12))/20+(1/104+1/298)*φ^2)/2)*2*π

  39. Paul Vaughan says:

    Tie.in de Rop 4 16 Figures D-light

    Review
    lunar draconic & anomalistic
    5.99685290323073 = beat(0.0754402464065708,0.0745030006844627)
    with anomalistic year
    1814.31362251033 = slip(5.99685290323073,1.00002638193018)

    aliquot sequence for B
    s(4370) = 4270
    s(4270) = 4658
    s(4658) = 2794
    s(2794) = 1814
    s(1814) = 910
    s(910) = 1106
    s(1106) = 814
    s(814) = 554
    s(554) = 280
    s(280) = 440
    ————————– see below
    s(440) = 640
    s(640) = 890
    s(890) = 730
    s(730) = 602
    s(602) = 454
    s(454) = 230
    s(230) = 202
    s(202) = 104
    s(104) = 106
    s(106) = 56
    s(56) = 64
    s(64) = 63
    s(63) = 41
    s(41) = 1
    s(1) = 0

    note:
    1106/2 = 553, 1106/2+554 = 1107, 4428 = 4*1107, & 744 = 640+104 demystify powers below the perfect level 28 4th power split of M & Leech.

    Also note 836 sequence overlap with B:
    s(836) = 844
    ————————–
    s(844) = 640
    s(640) = 890
    s(890) = 730
    s(730) = 602
    s(602) = 454
    s(454) = 230
    s(230) = 202
    s(202) = 104
    s(104) = 106
    s(106) = 56
    s(56) = 64
    s(64) = 63
    s(63) = 41
    s(41) = 1
    s(1) = 0

    Just imagine how much more we have to learn about number theory before we even have the basic aggregation criteria foundations needed to even BEGIN sensible climate stability exploration. Today’s politics are hopelessly intractable (ever since lockdowns became the west turn weapon of homeland financial terror). With monstrous leadership failures ALL across the west — weather left or right — George Polya’s advice is prescient and clear:

    Solve a simpler problem.

    flurry of miscellaneous puzzle pieces:

    s(652) = 49+149+298 = 496 = s(496) ——- perfect
    652/4 = 163 points to Ramanujan again.

    s(25) = 6 —– links to review: s(298) = 152
    s(6) = 6 ——- This is how you recognize perfect numbers.

    s(8128) = 8128 ——- perfect
    4428 = 8128 – 3700 = 4370 + 58 ———- 37 & 58 link precisely to 104

    UNtouchABull Branch

    de Rop fits into the aliquot sequence for the baby monster and 836 is a dead-end branch — what it means to be untouchableoff that.

    Arithmetic Dynamics
    =
    Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. […] A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
    =

    For an illustrative example, take a look at how simple this recipe is: […] “Mathematics may not be ready for such problems.” […] “is an extraordinarily difficult problem, completely out of reach of present day mathematics.”

  40. Paul Vaughan says:

    The Green Monster

    104.000044599386
    104.000034332275 = ⌊(e^√58π)^(1/p)⌉^p – e^√58π for p=2,4
    0.000009872218 = % error

    24.0679036739334
    24.067904774739 = ⌊(e^√7π)^(1/p)⌉^p – e^√7π for p=2,3,4,6,12
    -0.000004573749 = % error

    “don’t speak” — no. doubt

  41. Paul Vaughan says:

    SIM pull iff √(Φ-φ) U-N express sun:

    y = 24.067904774739 = ⌊(e^√7π)^(1/p)⌉^p – e^√7π for p=2,3,4,6,12
    x = 104.000034332275 = ⌊(e^√58π)^(1/p)⌉^p – e^√58π for p=2,4

    11.8626151191159=1/((6/y+(1/4)/x)/5+(1/x+1/298)*φ^2)
    11.8626151546089=1/J
    -0.000000299201=%error

    29.4474963123504=1/(1/x+1/298)/φ^2
    29.4474984673838=1/S
    -0.000007318222=%error

    19.8650369676306=5/(6/y+(1/4)/x)
    19.8650360864628=1/(J-S)
    0.000004435773=%error

    8.45614555057611=1/((6/y+(1/4)/x)/5+2*(1/x+1/298)*φ^2)
    8.4561457463176=1/(J+S)
    -0.000002314784=%error

    61.0464998394537=1/((6/y+(1/4)/x)/5-(1/x+1/298)*φ^2)
    61.0464822565173=1/(J-2S)
    0.000028802538=%error

  42. Paul Vaughan says:

    Perfect Fit

    19=s(77)
    37=s(323)

    28=(s(77)+s(323))/2=(19+37)/2

    19=28-9=s(77)
    37=28+9=s(323)

    496=298+49+149
    323=298+25=196883-196560

    196585=196883-298=196560+25
    s(298)=152=d(2,1/2,25)
    s(152)=148=s(d(2,1/2,25))

    248=149-49+148=496/2
    149=72+77
    77=152+248-323=248-171—- halve perfect mnEMonICm$y$nUSwanOvert(U-N)
    72=149-152-248+323=(323-152)-(248-149)=171-99 ————- m$y$nUSgreatsKEY
    171=323-152

    208 = 8128 – 7920 ———– M11
    s(8128) = 8128

    95 = 37 + 58 = 70 + 25 = √(73500/15) + 25 = s(5#)+s(28)+25 = 42+28+25
    s(95) = 25
    s(25) = 6
    s(6) = 6

    hitchhiker miss sol any us prime more real goal act IC guides?

    42 = (84)*(28) / (84 – 28) = (84)*(28)/((84+28)/2) — beat = harmonic mean
    = 70 – 28 = (210/2)*(70) / (210/2 + 70) = (7*5*3*2)*(70/2) / (7*5*3*2 – 70/2)

  43. Paul Vaughan says:

    Reviewing History with 2020 Hindsight

    Nature weaves convergent strands in perfect bundles. This line of communication was previously arrested by filter “misbehavior”.

    =
    the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions with the same number of terms.

    It is possible to interpret the Sylvester sequence as the result of a greedy algorithm for Egyptian fractions, that at each step chooses the smallest possible denominator that makes the partial sum of the series be less than one. Alternatively, the terms of the sequence after the first can be viewed as the denominators of the odd greedy expansion of 1/2.

    […] Curtiss (1922) describes an application of the closest approximations to one by k-term sums of unit fractions, in lower-bounding the number of divisors of any perfect number, and Miller (1919) uses the same property to upper bound the size of certain groups.
    =

    Sylvester’s sequence 2, 3, 7, 43, 1807, … can be viewed as generated by an infinite greedy expansion […] Truncating this sequence to k terms and forming the corresponding Egyptian fraction, e.g. (for k = 4)”

    1 = 1/2+1/3+1/7+1/43+1/2/3/7/43

    s(n) = aliquot sum of n
    s(4370) = 4270

    100 = 4370 – 4270 = 4370 – s(4370)

    70 = √(73500/15) — recall 73500 = 2*36750 key weighted harmonic mean from Laskar

    4200 = s(4370) – 70 = 4270 – 70
    4300 = 4370 – 70

    42 = (s(4370) – √(73500/15)) / (4370-s(4370))
    43 = (4370 – √(73500/15)) / (4370-s(4370))

    The first four of these numbers are one less than the corresponding numbers in Sylvester’s sequence, but then the two sequences diverge.

    1806 = 2*3*7*43 = 42*43
    = ( 4370*s(4370) – √(73500/15)*(4370+s(4370)) + 73500/15 ) / (4370-s(4370))^2

    271 = 496 – 15^2 = ⌊√73500⌉
    s(496) = 496

    489426 = 1806 * 271 ———– count.sol luck D-own sat U-N 4 XRview from fan-wise “monster” seats

  44. Paul Vaughan says:

    Rome and knew join general
    ….each let us group monde stir us sly stable parameter√(Φ-φ)shh!UN:

    d(p,r,k) = ⌊(e^π*k^r)^(1/p)⌉^p – e^π*k^r

    For example
    323.013383387384 = d(2,1/3,59) = ⌊(e^π*59^(1/3))^(1/2)⌉^2 – e^π*59^(1/3)
    and
    0.0133833873842377 = d(1,1/3,59) = ⌊(e^π*59^(1/3))^(1/1)⌉^1 – e^π*59^(1/3)
    such that
    323 = d(2,1/3,59) – d(1,1/3,59)

    Incidentally note us:
    323 = (59-12)*59*(59+12) – 196560 = 196883 – 196560 = 298 + 25

    Standby 4 perfect primorial leech tie.in.

  45. Paul Vaughan says:

    Monster US$green.in 1984=4*s(496)

    before rolling far enough away to allow Stuart to score.”

    =
    496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. […] Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case.
    […]
    E8 has real dimension 496.

    The number 496 is a very important number in superstring theory. In 1984 […] Green and […] Schwarz realized that one of the necessary conditions for a superstring theory to make sense is that the dimension of the gauge group of type I string theory must be 496. The group is therefore SO(32). Their discovery started the first superstring revolution. It was realized in 1985 that the heterotic string can admit another possible gauge group, namely E8 x E8.
    =
    https://en.wikipedia.org/wiki/496_(number)

    =
    There is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form […] of E8, and has an outer automorphism group of order 2 generated by complex conjugation.
    =
    https://en.wikipedia.org/wiki/E8_(mathematics)#Real_and_complex_forms

    248 = s(298)+96 = s(s(298))+(104+96)/2 = (2*72)+104 = (298-2*77)+104
    = (72+77)+49+(104+96)/4 = (149+49)+(149-49)/2 = s(496)/2

    5=77-72=harmonic mean of proper divisors of 496

    Perfect Moonshine

    “ayaM the jigsaw, man” — “more hue moon than WHO man” whyIT22oN?bei

    3 = √(absolute mean deviation from perfect) won over U-N see cure IT count.sol tip:

    s(28)=(s(323)+s(77))/2=(323+77)-s(298)
    ware p=2, r=1/2, & k=25 for U-N dr. read m$y$n US d(p,r,k)

  46. Paul Vaughan says:

    37 FEAT HI.in Left Fields φ niche-D “a11y” caught sum green monster:

    φ = 2*cos(s(71^2)/s(59^2)/s(47^2)*8*π) where s(n) = aliquot sum for n
    φ = 2*cos(s((59+12)^2)/s(59^2)/s((59-12)^2)*8*π)

    196883 = 71*59*47 = (59+12)*59*(59-12)

    Naive weather left or right, IT’s AI “just” a Que. when so dense.
    Don’t Mayan D-proof: “just” go ON with Bei=lief.in 2+2=5 no. when 4*s(496) = 1984 is AB US sov. “green” morse un out field doors vic[tory] Dev.O[s] lie low.

    “[…] a popular target for right-handed hitters […] the Green Monster was not painted green until 1947 […] Yellow numbers are used to represent in-inning scores […] left-field distortion is offset by the odd shape and generous size of right field […] The placement of the ladder is noteworthy given the fact that it is in fair territory […] a high fly that ricocheted first off the ladder, and then the head of outfielder [___]”

    323 = 71*59*47 – 196560
    s(323) = 37

    “The Coke Bottles on the left light tower were a target for power-hitters […]”

    104 ~= ⌊(e^√37π)^(1/2)⌉^2 – e^√37π

  47. Paul Vaughan says:

    Bei. U-N D!Superst[r]ing

    Reverse-eng.in.ear.ring oldschool “luminary” assembly with 2020 hindsight, we find attached a perfect leech:

    490000 = 15*70^2 / asin(φ/2) * 2π

    Lol! What curry US IT didn’t even (“don’t speak” no. doubt) tell us. Tech no. CR at IC govern ants?ware.in monde stir US $ sly “green”.

    Sylvester’s Sequence (“more hue moon than WHO man” = whyITzoombei) goes as far as possible past the perfect leach, account.in 4 minor deviations from superstring perfectshhU-N.

    1 / 164.791265692394 = (2/1806-1/3)J+(2/1806+1)S-2/1806/15/70^2*asin(φ/2)/π+1/3/s(4370)
    -0.000030309658 = % error

    1 / 84.0168557977672 = 2*((2/1806-1/3)J+(2/1806+1)S-2/1806/15/70^2*asin(φ/2)/π)-1/3/s(4370)
    0.000011754317 = % error

    peer√(Φ-φ)ed dub e/11 ln acts pawn ants yell

    1 / 164.791269677405 = (2/1806-1/3)J+(2/1806+1)S-2/1806/1806/271+1/3/s(4370)
    -0.000027891441 = % error

    1 / 84.0168578694536 = 2*((2/1806-1/3)J+(2/1806+1)S-2/1806/1806/271)-1/3/s(4370)
    0.000014220116 = % error

    4270 = s(4370) gives the Bay’s first-step B-estimate for U & N as does 4428 = 4370+58 = 8128-3700 for J & S.

    “on my blue PR(INT) IT’SIM F(U-N) IC; spin that record B — Lady G “just” D-ants

    Anchoring thus contrasts oldschool “green” leach attached to mainstream mindset verse USSylvester’s double exponential greed sequence.

    The 1st pair of U & N estimates above is based on J & S estimates.
    The 2nd pair of U & N estimates above is based on J & S from Seidelmann (1992).

    FormAI pol!shh, IT’s missUN no linksnow:

    This is a diagnostic strategy precisely exposing an exponential in clear conflict with a double exponential. A hair-splitting subtle difference in this case is period-doubling.

  48. Paul Vaughan says:

    Scene IC Route to Secure IT Count Sol

    4370 = 2*5*19*23

    “√(Φ-φ) lost my keys
    √(Φ-φ) can’t C strait anymore
    What’s go.in.on.on the floor?
    Spin that record Babe” — Lady Gaga Just D-ants

    Φ(2) = 1
    Φ(5) = 4
    Φ(19) = 18
    Φ(23) = 22

    Euler’s totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n). This function gives the order of the multiplicative group of integers modulo n […]”

    Φ(3) = 2
    Φ(37) = 36

    s(4370) = 4270
    “Then one foggy Christmas eve…” — Rue Dolph.in Red Knows Reign D-a√(Φ-φ)r
    5106 = 4270 + 836 = 2*3*23*37

  49. Paul Vaughan says:

    1584 Miscellaneous Tips from General Ramanujan’s Travel Guide

    Today’s trip begins with ET (Euler’s Totient).

    count: 40
    lowest:
    Φ(1679) = 1584
    the next 2:
    Φ(1691) = 1584
    Φ(1985) = 1584
    double:
    Φ(3382) = 1584
    Φ(3970) = 1584
    highest:
    Φ(6210) = 1584

    Having boldly left a little mystery, we rightly move on not icing hawk key:

    Φ(4370) = 1584
    Φ(5106) = 1584

    It is possible for an infinite set of integers to be pairwise coprime. Notable examples include the set of all prime numbers, the set of elements in Sylvester’s sequence, and the set of all Fermat numbers.

    Φ(4270+836) = 1584
    Φ(s(4370)+836) = 1584

    Recall that 836 is untouchable, meaning it’s not an s(n) for any n — i.e. there is NO number with aliquot sum = 836.

    A comparative view of Jovian order is shaping up:
    global exponential constraint with no local adjustment
    • global order subtly-retuned (prime-D-iffew prefer) 4 fit with double-exponential local greed

    104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π = 156816^2 – e^√58π
    104.001742574386 = ⌊(e^√22π)^(1/2)⌉^2 – e^√22π = 1584^2 – e^√22π
    104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π = 396^4 – e^√58π

    396 = √156816
    1584 / 396 = 4 = 396 / 99
    156816 / 1584 = 99
    √(58-22) = √36 = 6 = s(6) = s(25)

    d(p,r,k)
    d = difference
    p = power
    r = root
    k = count or level in docks
    152 = d(2,1/2,25) – d(1,1/2,25)
    152 = (d(3,1/2,25) – d(1,1/2,25)-8744)/2 = (9048-8744)/2 = 304/2

    review:
    levels k=10,13,18,22 converge on 104
    quadruple those
    levels k=40,52,72,88 converge on 8744

    4370 = (84*104+8744)/4
    We explore these things in stages. Each stage brings more clarity.

    Landscape ecology hierarchy theory and years of field experience using detailed botany taxonomy keys helped base a mindset to orient in rich territory.

    Conventional linear “thinking” about what’s “best” doesn’t help see (sort and classify) how every thing naturally fits in God’s creative design.

    s((84*104+8744)/4) = 4270
    Φ(s((84*104+8744)/4)+836) = 1584

    Discrete-continuous relations are full of nested, recursive, hierarchical structure. Western math education is fatally deficient. What do you think bad elite (to be neither confused nor conflated with good elite, if such a thing exists) can do with ignorance of trade secrets?

    Given a chance to speak (remember what Conway & Norton said) to the pope about luck D-own west turn spread of severe financial terror past just “rogue” nations to “count less” millions of homeland western citizens, what might I say?

    “and if(IT)s reel then √(Φ-φ) don’t want tune O
    √(Φ-φ) no. ya reel good”
    — “don’t speak” no. doubt

    139560 = d(3,1/2,58) – d(1,1/2,58)
    70^2 = (196560-139560-8744+744)/10 = 73500/15 = 2*36750/15
    review: 744 = d(1,1/2,k) – d(3,1/2,k) for k = 43-24, 43, 43+24

    836 is the next-lowest weird number after 70. 836 is untouchable. 70 is not. “70p is weird for all primes p ≥ 149

    A ware no. word scan do, weave numbers.

  50. Paul Vaughan says:

    CorrectS[hh!]UN[8]O[R]well.in.sight

    The tech “know”so-called CR at IC “eXpeRts” never advise-D of period-dub e ln SIM MET try.

    “the bloop ill O-pens euRise
    is theRe a bet te[a]R weigh?
    a knew Religion pRess scRibed
    to those without the faith
    is IT 2 late 2 go back?
    is IT 2 late 2 go?
    theRe’s No. won heaR
    and peep e/11 eveRy waRe:
    yeaR .oN euR O-won” — Queens of the Stone Age

    Won step back.
    73500 = 3*(196883-4270/5-29)/8
    2 steps 4 word.

    489425.981694385 = 1 / ( -300J+904S-301U-301N )
    489426 = 2*3*7*43*271 = 1806*271
    -0.000003740221 = % error

    where ABCD estimate JSUN using Ramanujan’s Perfect Superstring:
    245061.761049564 = 1 / ( -300A+904B-301C-301D )
    245061.75 = (10*196560-4270-836)/8
    0.000004508890 = % error

    Recall: s(4370) = 4270

    First-order model’s base-D on the baby monster.
    Leech precisely IDs the point of symmetry that’s well-corrected by Sylvester’s double-exponential greed.

    We’ll take the seen IC route away from luck D-own weather Joe Trump or Don Biden.
    1 = 1/2 + 1/2 be 4+after USelection: know matter what curry US IT obstructs, free dem. from eur. increasingly savage financial terror schemes directed not only at “rogue” nations abroad but “count less” millions of homeland western citizens left unable to pay rent and buy food since they were CANCELED in a west turn tournament left by incompetent organizers with no consolation round.

    PC’s plan’s UNstable. Withdrawn BE support can only be restored under more favorable circumstances should they stably develop to sustain trust — including: no more Boris $ under mind “free” doom for a monarchy planning a reign of widespread homeland financial torment with no consolation.

  51. Paul Vaughan says:

    Ramanujan’s the man who knew in.ph.in.IT’s perfect superstring:

    22.1392314983836 = 1/(3V-5E+2J)
    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    11 = d(2,1/2,6) – d(1,1/2,6) = 59mod24 = Φ(71mod24)/2 = Φ(47mod24)/2 = Φ(23)/2
    6 = s(6) = s(25) = Φ(7) = Φ(9) = Φ(14) = Φ(18) = Φ(Φ(19))
    6 = Φ(s(2*Φ(Φ(22)))) = Φ(s(s(s(s(22))))) = Φ(s(s(Φ(22))))
    6 = Φ(s(2*Φ(Φ(Φ(23))))) = Φ(s(s(s(s(Φ(23)))))) = Φ(s(s(Φ(Φ(23)))))
    Φ(23)
    83.9908695437061 = Φ( d(4,1/2,6) – d(1,1/2,6) )/2 – d(1,1/2,6)
    83.9908202014941 = 1/(J^2-S^2)/2
    104 = d(1,1/2,s(d(4,1/2,6)-d(1,1/2,6)))-d(2,1/2,s(d(4,1/2,6)-d(1,1/2,6)))

    Note We/11:
    lim s→∞
    (φ^(2s)+1/s)^(e/s+1/(2s))
    = φ(φφ)^e

  52. Paul Vaughan says:

    Can’s e/11-D: mmmIC Dawn a11-D’s Perfect Hire Arch IC AI Notice

    225 = d(1,1/2,9) – d(3,1/2,9) = 15^2
    71 = d(1,1/2,9) – d(2,1/2,9)
    Φ(71) = 70
    70^2 = 1^2+2^2+3^2+…+22^2+23^2+24^2

    73500 = ( d(1,1/2,9) – d(3,1/2,9) )^(1/2)*Φ( d(1,1/2,9) – d(2,1/2,9) )^2
    36750 = ( d(1,1/2,9) – d(3,1/2,9) )^(1/2)*Φ( d(1,1/2,9) – d(2,1/2,9) )^2/2

    “√(Φ-φ)awe sum things√(Φ-φ)thought.in√(Φ-φ)verse awe
    EU’ve gotta whole.in U.in.N√(Φ-φ)verse awe”
    — Queens of the Stone Age “Monster’s.in the Paris Sol”

    18 = d(2,1/5,Φ(Φ(163))) – d(1,1/5,Φ(Φ(163))) = Φ(Φ(Φ(163))) = Φ(19)
    22 = Φ(Φ( d(1,1/5,Φ(Φ(163))) – d(5,1/5,Φ(Φ(163))) )) = Φ(23)

    47 = d(1,1/5,Φ(Φ(163))) – d(5,1/5,Φ(Φ(163)))
    59 = 2*d(1,1/5,Φ(Φ(163))) – d(3,1/5,Φ(Φ(163))) – d(5,1/5,Φ(Φ(163)))
    71 = d(1,1/5,Φ(Φ(163))) – d(3,1/5,Φ(Φ(163))) = s(s(Φ(163)))

    225 = d(4,1/5,Φ(Φ(163))) – d(1,1/5,Φ(Φ(163)))
    342 = d(1,1/5,Φ(Φ(163))) – d(6,1/5,Φ(Φ(163))) = 2*171

    73500 = ( d(4,1/5,Φ(Φ(163))) – d(1,1/5,Φ(Φ(163))) )^2 * Φ( d(1,1/5,Φ(Φ(163))) – d(3,1/5,Φ(Φ(163))) )^2
    36750 = ( d(4,1/5,Φ(Φ(163))) – d(1,1/5,Φ(Φ(163))) )^2 * Φ( d(1,1/5,Φ(Φ(163))) – d(3,1/5,Φ(Φ(163))) )^2/2

    Φ(Φ( 2*d(1,1/5,Φ(Φ(163))) – d(3,1/5,Φ(Φ(163))) – d(5,1/5,Φ(Φ(163))) )) =
    28 = s(28) = Φ(58) = Φ(Φ(59))
    6 = s(6) = Φ(18) = Φ(Φ(54)) = Φ(Φ(Φ(162))) = Φ(Φ(Φ(Φ(163)))) = Φ(Φ(19))

  53. Paul Vaughan says:

    IC a few obviOus typos: IT’s no. thing per sun AI on the C[ENSO]Rship.

  54. Paul Vaughan says:

    In Julian years:

    1.00001743371442 = 1/E

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)

    *365.25 gives days:

    365.256367733331 = 1/E
    365.256367664193
    0.000000018929 = % error
    That’s 0.597351554461056 seconds per century.

  55. Paul Vaughan says:

    1.61803398874989 = φ
    0.618033988749895 = Φ

    1 = φ – Φ
    Φ = 1 / φ

    2.71828182845905 = e

    2.71828182845905 = (1-Φ^1)^(0/1)/(1-Φ^2)^(2/2)/(1-Φ^3)^(3/3)/(1-Φ^4)^(2/4)/(1-Φ^5)^(5/5)/(1-Φ^6)^(1/6)/(1-Φ^7)^(7/7)/(1-Φ^8)^(4/8)/(1-Φ^9)^(6/9)/(1-Φ^10)^(3/10)/(1-Φ^11)^(11/11)/(1-Φ^12)^(4/12)/(1-Φ^13)^(13/13)/(1-Φ^14)^(5/14)/(1-Φ^15)^(7/15)/(1-Φ^16)^(8/16)/(1-Φ^17)^(17/17)/(1-Φ^18)^(6/18)/(1-Φ^19)^(19/19)/(1-Φ^20)^(8/20)/(1-Φ^21)^(11/21)/(1-Φ^22)^(9/22)/(1-Φ^23)^(23/23)/(1-Φ^24)^(8/24)/(1-Φ^25)^(20/25)/(1-Φ^26)^(11/26)/(1-Φ^27)^(18/27)/(1-Φ^28)^(12/28)/(1-Φ^29)^(29/29)/(1-Φ^30)^(9/30)/(1-Φ^31)^(31/31)/(1-Φ^32)^(16/32)/(1-Φ^33)^(19/33)/(1-Φ^34)^(15/34)/(1-Φ^35)^(23/35)/(1-Φ^36)^(12/36)/(1-Φ^37)^(37/37)/(1-Φ^38)^(17/38)/(1-Φ^39)^(23/39)/(1-Φ^40)^(16/40)/(1-Φ^41)^(41/41)/(1-Φ^42)^(13/42)/(1-Φ^43)^(43/43)/(1-Φ^44)^(20/44)/(1-Φ^45)^(24/45)/(1-Φ^46)^(21/46)/(1-Φ^47)^(47/47)/(1-Φ^48)^(16/48)/(1-Φ^49)^(42/49)/(1-Φ^50)^(20/50)/(1-Φ^51)^(31/51)/(1-Φ^52)^(24/52)/(1-Φ^53)^(53/53)/(1-Φ^54)^(18/54)/(1-Φ^55)^(39/55)/(1-Φ^56)^(24/56)/(1-Φ^57)^(35/57)/(1-Φ^58)^(27/58)/(1-Φ^59)^(59/59)/(1-Φ^60)^(16/60)/(1-Φ^61)^(61/61)/(1-Φ^62)^(29/62)/(1-Φ^63)^(36/63)/(1-Φ^64)^(32/64)/(1-Φ^65)^(47/65)/(1-Φ^66)^(21/66)/(1-Φ^67)^(67/67)/(1-Φ^68)^(32/68)/(1-Φ^69)^(43/69)/(1-Φ^70)^(25/70)/(1-Φ^71)^(71/71)/(1-Φ^72)^(24/72)/(1-Φ^73)^(73/73)/(1-Φ^74)^(35/74)/(1-Φ^75)^(40/75)

  56. Paul Vaughan says:

    Moderators: Goal done approximation caught in the filter natural e starts on the left, extending infinitely far right to phinally reach strict mathematical proof.

  57. Paul Vaughan says:

    Perfect He’er

    M the orrery: 496 = s(496) = 2/3*744
    He’er.ET.IC string theory comm. pact iff IC k? shh! UN.

    Knock, knock.
    WHO’s there?
    J.in.very.ant.
    Jane VI rant WHO??
    j-invariant 2pi/3 is anyon nome?

    194 = average(225,163)
    172 = 194-22 = 9+163
    171 = 9+ET(163)

  58. tallbloke says:

    Woah! There’s been a lot of work going on in here.

  59. Paul Vaughan says:

    Rog, the tide was flooding as I paddled across the narrows.
    A long, black helicopter flew over very low, commanding attention.
    After it passed, something falling on what appeared to be a small parachute caught my eye.
    I watched the gentle landing on the water’s surface.
    I altered course to inspect.
    It was not a parachute but a pink heart-shaped helium balloon, with a draping note including the word “fun” and a black pentagon with points numbered 1 to 5 in white.
    Δ(√(73500/15)) = Δ(70) = 25 = 5^2

  60. Paul Vaughan says:

    Notation Alert

    I’m refining notation — going back to R() for Ramanujan and d() is the difference of a pair of R()s that gives a positive integer. Power p=1 (the fraction) is always a member of the differencing pair.

    I’ll illustrate with an example:

    R(1,1/5,54) = -0.00832631991693233 = ⌊(e^π*54^(1/5))^(1/1)⌉^1 – e^π*54^(1/5)

    R(3,1/5,54) = -71.0083263199169 = ⌊(e^π*54^(1/5))^(1/3)⌉^3 – e^π*54^(1/5)
    R(5,1/5,54) = -47.0083263199169 = ⌊(e^π*54^(1/5))^(1/5)⌉^5 – e^π*54^(1/5)

    71 = d(3,1/5,54) = R(1,1/5,54) – R(3,1/5,54)
    47 = d(5,1/5,54) = R(1,1/5,54) – R(5,1/5,54)

    59 = average(d(3,1/5,54),d(5,1/5,54))

    The convention change makes notes more terse.

  61. Paul Vaughan says:

    Juncture 162

    With UN a poll lag(ET)IC mnemonic PRloom.inairai$e We beg.in IT’s j(U-N)k shh!invaryant.

    At a juncture deciding weather a new day dawns, recalling Polya’s prescient advice to solve a simpler problem, let’s simply consider:

    1. inverse totients (upscale branches).
    2. totient sequences (downscale stem shared by upscale branches).

    What cross-diss up plan eerie curry O’s IT?

    There’s a cureOus lack of conventional mainstream reference to totient sequences. IT’s the “tie UN” USelection 4 “tie = won” if the method of loci (mind palace memory method) works well enough.

    Wikipedia links to the concept aliquot sequence. The concept is used to define perfect numbers. Recall that n is perfect if n = s(n). For example, a chain we’ve noted:

    s(652) = 496 = s(496) = 49+149+298 = (2/3)*744

    652 = 4*163 gives a path to 496, meaning 496 is touchable (what they call it). Once the stem reaches 496, it ends. Perfect numbers are aliquot sequence termini.

    Notably 28 is untouchable. If you start an aliquot sequence at 28 it stays there (i.e. 28 is perfect), but since 28’s untouchable you can’t use any aliquot sequence from anywhere else to get to 28.

    The analogies for aliquot’s touchable and untouchable in ET (Euler Totient) context are totient and nontotient. We’re building a thesaurus to help us navigate cross-disciplinary communication.

    Having appreciated the fundamental utility of aliquot sequences as a sorting and classification guide, I quite naturally, instinctively, and enthusiastically catalogued ET sequences and began looking for analogies. When I write notes I’m calling the ET sequences totient sequences. As you’ll see they’re a moonlight key to monstrous sorting and classification.

    There’s UN other type O seek wins I’m explore ring 4 EU tell IT as a fun dem. meant AI sort & class iff IC case sun guide, D-note-D by the small greek letter D-e/11-TA, but 4 now let’s just note key junction 162:

    Φ(163) = 162 —– 163 is prime, nontotient, and (TAKE NOTE) the largest of the 9 Heegner numbers
    Φ(243) = 162 = Φ(3^5) ———– 243 is nontotient
    Φ(326) = 162 = Φ(2*163) ———– 326 is nontotient
    Φ(486) = 162 = Φ(2*3^5) —– 486 = 2*243 is totient (points up for example to 729 = 3^3^2)
    ^ upscale ^
    ————–
    v downscale v
    Φ(163) = 162
    Φ(162) = 54
    Φ(54) = 18
    Φ(18) = 6
    Φ(6) = 2
    Φ(2) = 1
    Φ(1) = 1

    Discrete functions I developed (generalizing a now infinitely less “mysterious” Ramanujan “trick”) IDed almost-integers as spikes that sort by modulus. Now I’m tracing spike origins to orient in conventional paradigms.

    Context: Here we’re off in a corner focused on e and pi (special values assigned to 2 variables), but stay aware that the method generalizes several-fold.

    Note the 54 in the downscale totient chain above. By now some readers may be thinking: “There are so many numbers. How can I keep track of them all? How can so many be important?” Suggestion: As with specialized members playing different roles on a team, big-picture fans note well where and how pieces fit the whole. We can see de Vries in discrete context without which de Vries mnemonICally wood knot B de Vries.

    Let’s note 1 more branch to 54:

    Φ(81) = 54 = Φ(162/2)
    Φ(54) = 18
    Φ(18) = 6
    Φ(6) = 2

    Note that the branches above 54 from 162 & 81 (half of 162) share a common stem from 54 down.

    I write left neither 4 USelection rig nor well divided audience no. win next comm. on C[ENSO]Rship (weather easy or not) will hinge on “Junk Sure” Φ(163).

    God bless you all — weather left, right, Chinese, Russian, North Korean, Persian, American, OR CZECH (U. Charles!) — as with General Ramanujan We D-tour threw monstrous moonshine on an otherwise hazard US trail.

    Here I leave 1 mysterious note:
    Φ(4374) = 1458

  62. Paul Vaughan says:

    Won Course Braiding Streams from Left and Right

    This comment will be left deliberately terse to rightly invite nonpartisan appreciation of nature.

    M rep review:
    196883 = 47*59*71 = (59-12)*(59)*(59+12)

    Φ(118) = 58 —————————– 118 is nontotient (means no branches upward from there)
    Φ(58) = 28 = s(28) = Φ(59) = Φ(29) ———– 29 & 59 nontotient
    Φ(28) = 12
    Φ(12) = 4
    Φ(4) = 2
    Φ(2) = 1
    Φ(1) = 1

    Recall 28 = s(28) means 28 is “perfect” (in number theory lingo).

    We’ve looked at totient sequences passing through 54 & 58. Here’s the intermediate chain passing through 56 = average(58,54) = 2*28 = 2*s(28) :

    Φ(236) = 116 = Φ(177) = Φ(354) — 177, 236, 354 nontotient (meaning capped — i.e…)
    Φ(116) = 56 = Φ(87) = Φ(174) — 87, 174 nontotient (no branches up from stem below go higher)
    Φ(56) = 24
    Φ(24) = 8
    Φ(8) = 4
    Φ(4) = 2
    Φ(2) = 1
    Φ(1) = 1

    Summary

    163 – 59 = 104 IDs del pezzo surface structure.
    A del Pezzo surf ace circuit starts and ends with 59:

    Φ(59) = 58
    d(2,1/2,58)=d(4,1/2,58)=104
    163=59+104
    Φ(163) = 162
    Φ(162) = 54
    59=average(d(3,1/5,54),d(5,1/5,54))=average(71,47)

    The perfect connection:
    Φ(58) = 28 = s(28) = average(58,54)/2

  63. Paul Vaughan says:

    ZZ$ox: Sharp Dressed Mayan$Top

    Note well Heegner M-ET del PeZZo surf ace on the C[ENSO]Rship:

    E6 (degree 3): 28 = 27+(6-5) = 27+(4-3) = 27+1 (line)
    E7 (degree 2): 58 = 56+(7-5) = 56+(4-2) = 56+2 (bitangent)
    E8 (degree 1): 243 = 240+(8-5) = 240+(4-1) = 240+3 (tritangent)

    the mob comm.s.CR.awe.11.in/out — Queens of the Stone Age

    My[UN]key orientsuper string he’er[ET(IC)]: Table 1 p.19.

    1728 = 2*744+240
    8128 = 7920+208 = s(8128) —- 8128 is perfect; 7920 is M11 order

    Monde stir seats high above left Field’shhall O[We/11]play in boss tune: my M?(y/n)-dis up hear on UNother wave, cove.ReD.in he’er” — Queens of the Stone Age “Monsters.in the Paris Sol.”

  64. Paul Vaughan says:

    Ally Caught Up in Bushy Taled Top O Log IC AI Totient Trees

    So far weave no.comm.on.tory.on the 3rd root, which first sprouts 59 trees in a maze sun 4 est con text:

    59 = average(47,71)
    196883 = 47*59*71 = (59-12)*(59)*(59+12) ————– M rep
    196560 ———————————————————— Leech rep

    71 = s(s(162)) = s(s(Φ(163)))

    323 = d(2,1/3,59) = 196883-196560 = 298 + Δ(70)
    s(323) = 37
    d(2,1/2,37) = 104

    493 = d(3,1/3,59)
    s(493) = 47

    At canopy level won squirrel isn’t lost in academic publication floods algebra storm.

    Φ(59) = 58
    d(2,1/2,58) = d(4,1/2,58) = 104

    Φ(58) = 28 ————- remember 28 = s(28) means 28’s perfect
    d(4,1/2,28) = 196585 ——- perfect number 28 gives a perfect split
    298 = 196883-196585 = 323 – Δ(70)
    25 = 196585-196560 = Δ(70) = Φ(70) – μ(70) = 323 – 298

    de vries first order quantum hale effect jovian circuit anyon?
    7920+208=8128 ——– 7920 is M11 order; 8128 is perfect
    8128-3700=4428 —— 37 is sharp 104-yielding level
    4428-58=4370 ———- 58 is sharp 104-yielding level; 4428 defines JS relations to first order
    4370=2*5*19*23 ——– s(4370) = 4270 ties UN to JS (first order); 4370 is B rep
    7920=5*Φ(5)*Φ(19)*Φ(23) —— yet another bilateral-pentagonal 5/2 key ET IDs

    Chased by owls while looking for food and shelter just watch an amazed squirrel run through the circuit tree.

  65. Paul Vaughan says:

    Route 5 Steps to Fifth Root.in Sight

    N8O worry O[r] left.in on dawns golf righter keys:

    “V(ic)tory i$ Mayan — Putt, tee $myth “The War e ^ O[r] ”

    Polya: “Solve a simpler problem.”

    66, 71, 76 —– 2*76 = 152 = 128+24
    54, 59, 64 —– 2*64 = 128 = average(152,104) = 152-24 = 104+24
    42, 47, 52 —– 2*52 = 104 = 128-24

    Phew! C[eNSO]Rship nearly had US divided and conquered ITself! be 4 share ring nonpartisan finds comm. on! ground.

    That was the look forward to a simple ending.
    Now let’s back track.

    54 is the big fifth-root spike.
    47 & 71 are 2 of the noteworthy products.

    Their average 59 is the big third-root spike.
    Let’s simply explore fifth-root connect sun with square-root.

    59-5 = 54 = s(42)
    59+5 = 64

    71+5 = 76
    2*76 = 152

    47+5 = 52
    2*52 = 104

    (71+5)+(47+5) = 128 = 2*(59+5)
    152+104 = 256 = 4*64

    47-5 = 42
    256+42 = 298 ————– JS first order (review)

    A Maze UN: D-air Rat “Miss Fear” IC Look Do[w]n

    Something te/11s US this isn’t what climb ET “super” comm. pew dare sore luck kin fear.

    fall O mystery O jung11chi11ed…

    Caught “ally”: wood knot no. by curry US IT what’s UN known?

    s(42) = 54 ———- 42,
    s(54) = 66 ———- 54, &
    s(66) = 78 ———- 66 on the same branch share ring aliquot sea stem
    s(78) = 90
    s(90) = 144
    s(144) = 259
    s(259) = 45
    s(45) = 33
    s(33) = 15
    s(15) = 9
    s(9) = 4
    s(4) = 3
    s(3) = 1
    s(1) = 0

    We a11 no. IT‘s knot right 2 look left.in 2 blew collar find $UN ants air red quest yen.

    Putt turn wreck cog niche$yen.in phi sox red: “green” monde stir B(est) tune f(UN) way.

    “WHO’s the hunter. Whose the game?”

    s(5#) = 42 ——————— antswore: BUNT! (left field shh! all O)

    “I feel the beat call Eur. name
    I haled EU close.in victory
    I don’t want too tame Eur. animal style
    EU won’t be caged.in the call of the will.ed” — Patty Smyth “The Warrior”

    Red door well note won mystery 66 left unadduressed 4 coverage UN[he!ll]O-there day with $O asin(O[r]We’ll).

  66. Paul Vaughan says:

    Hale 240

    2, 12, & 240 are both highly and sparsely totient.

    1/H = 1/(3V-5E+2J) = 22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    Solving for 1/V:
    1/V = 1/((1/3)H+(5/3)E-(2/3)J) ———————— 1/J
    0.615197263582188
    0.615197263396975 = 1/V (Seidelmann 1992)
    0.000000030106 = % error

    lim s→∞
    (φ^(2s)+1/s)^(e/s+1/(2s))
    = φ(φφ)^e

  67. Paul Vaughan says:

    e-Luke Shh(Own) Guard Do[w]ns Perfect Links !Sol Look! Shone

    27, 81, & 243 are perfect totient numbers:

    ΣΦ(54) = ΣΦ(27) = 27 = 18+6+2+1
    ΣΦ(162) = ΣΦ(81) = 81 = 54+18+6+2+1
    ΣΦ(163) = ΣΦ(243) = 243 = 162+54+18+6+2+1

    https://oeis.org/A082897

    Φ(163) = 162 = Φ(243)
    Φ(162) = 54 = Φ(81)
    Φ(54) = 18 = Φ(27)
    Φ(18) = 6
    Φ(6) = 2
    Φ(2) = 1

    163-104=59
    Φ(59) = 58
    Φ(58) = 28 = Φ(29)
    Φ(28) = 12
    Φ(12) = 4
    Φ(4) = 2
    Φ(2) = 1

    Hale O Monster USlink B(ET) we’en 2 stems:
    104 = d(2,1/2,s(196883-196560)) = d(2,1/2,58) = d(4,1/2,58)

    Just little sure pries: All Lie !Caught! Hide U-N Truth

    “Sometimes the faster IT gets, the less EU need tune OTHseasurfacetemperatch[ET]O[r]well

  68. Paul Vaughan says:

    B(est) Tune Perfect Green CO2shh!UN Monster Level 7 in f(U-N weigh)

    φ(8128) = 4096
    φ(4096) = 2048
    φ(2048) = 1024
    φ(1024) = 512
    φ(512) = 256 = φ(496)
    φ(256) = 128
    φ(128) = 64
    φ(64) = 32
    φ(32) = 16 = φ(28)
    φ(16) = 8
    φ(8) = 4
    φ(4) = 2
    φ(2) = 1
    φ(1) = 0

    R(k,1/2,7) = 24.067904774739 = ⌊(e^√7π)^(1/k)⌉^k – e^√7π for k=2,3,4,6,12
    Calculating ⌊(e^√7π)^(1/k)⌉^k :
    k=2: 64^2 = 4096 = φ(8128)
    k=3: 16^3 = 4096 = φ(8128)
    k=4: 8^4 = 4096 = φ(8128)
    k=6: 4^6 = 4096 = φ(8128)
    k=12: 2^12 = 4096 = φ(8128)
    Notice:
    R(k,1/2,7) = 24.067904774739 = φ(8128) – e^√7π for k=2,3,4,6,12

    Again recall 28=s(28), 496=s(496), & 8128=s(8128) are perfect numbers.

    Best tune crash-course threw number theory seas solar system order UN dare green monster in fun ways left field.

    “Mathematics is the queen of the sciences—and number theory is the queen of mathematics.” — Gauss

    Left field creatures: Nature’s beauties there to sea (surf ace stem peer ashore) if Eur. will.int(el)look past the C[e^(NSO)]Rship. (IT AImost divided and conquered US.)

  69. Paul Vaughan says:

    Room O[r]s of short circuit in.wash.un.tune

    Mayan D-Palace readers remember SUM thing CO2shh!UN reversing the tale end of bollinger’s symbol IC 1952 pentagon vis IT. From “ok” LA home a rumor doesn’t eve UN have to be true or well 4 students 2 remember? The tale ends with a baby.

    You may 4 get 2 individuals, but not how they add up.in education AI controversy.

    Φ(69) = 44
    φ(69) = 25

    Political junkies love D-bait.in rumors weather true or not.

    Definition of cototient:
    Φ(n)+φ(n)=n

    Bollinger’s paper about a 44 year cycle was published in 1952 when the Pentagon’s focus was the Korean War. Let’s look at the pieces.

    The totient of 19 is 18.
    52 is half of 104 and written backwards 5^2 = 25.
    Half of 44 is 22 and that is the totient of 23.

    4370 = 19*5*2*23

    Script-writers, Bollinger’s theme song is Lady Gaga’s “Just D-ants”.
    Ally caught 4370 is 4270, pointing straight to U-N.

    Even worse, IT’s reverse land.in the river times galaxy:

    Take the 9 out of 1952. You have 152. Use the 9 with half of 22 (as did Ramanujan) to key Figure$16 tames 9*11=IX*XI 4×4 times just Rome UN no. morals.

    Of course Ramanujan knew 1584 is the totient of 4370 and every hitchhiker nos. 42+104+152=298 phinally ties US to jovian Hale weather through Junction 162 from “Ramanujan’s constant” to totient 58 via 59 or scaled perfectly threw ally caught 37=s((59-12)*59*(59+12)-196560) in 11 each lettuce.

    Really, is there any point in B-labor.in UNcloaked narrative any further at this point? That’s enough creative writing room or wells.in wash.un.tune. AIron court.in MSM D-baits key word door.in monster US KOre[]on wa[B] peek r&B baby [r]oomer fall O-win pan tug gone 69 CO2$UN symbol IC a11y !caught!

    Nons[ENS]e^Poll[O]see Puttin’ on monster US “green” links? ITees the treat e We sign-D.

  70. Paul Vaughan says:

    Five

    Constant new tettitory is a costly source of cross-disciplinary delight.

    I’ve decided number theory should be a staple of math education for all fields. It wasn’t a part of any program I ever studied, so I’m learning as I explore.

    Here I leave another note knowing specialists should be able to move much faster with general orientation.

    φ(70) = 46
    φ(46) = 24 = Φ(70) = φ(φ(70))

    70’s totient Φ(70) equals its double-cototient φ(φ(70)). That wasn’t covered in my ecology courses, but I now know its relevant to Earth’s stability.

    √Δ(70) = √(Φ(70)-μ(70)) = √(φ(φ(70))-μ(70)) = φ(25) = 5
    Δ(70) = Φ(70)-μ(70) = φ(φ(70))-μ(70) = φ(25)^2 = 5^2 = 25

    70 = √(73500/15)

    5 = √Δ(√(73500/15)) = √(Φ(√(73500/15))-μ(√(73500/15))) = √(φ(φ(√(73500/15)))-μ(√(73500/15)))
    = φ(Δ(√(73500/15))) = φ(Φ(√(73500/15))-μ(√(73500/15))) = φ(φ(φ(√(73500/15)))-μ(√(73500/15)))

    Alert readers: Watch for typos.

    Would naive hobbyists looking at Titius-Bode “Law” focus on how modular forms fit together stably? With 2020 hindsight our observations suggest no at step 1.

  71. Paul Vaughan says:

    Review and Consolidate

    This note gives technical clarification where there’s considerable risk of misunderstanding (below) after further elucidating a “perfect totient” structure introduced above.

    Notation: Φ() for ET and φ() for cototient.
    A subset of perfect totient stems alternates or zippers perfectly with cototient stems.

    Φ(729) = 486 = 2*φ(729)
    Φ(486) = 162 = 2*φ(243)
    Φ(162) = 54 = 2*φ(81)
    Φ(54) = 18 = 2*φ(27)
    Φ(18) = 6 = 2*φ(9)
    Φ(6) = 2 = 2*φ(3)
    Φ(2) = 1
    Φ(1) = 1

    φ(729) = 243 = Φ(729)/2
    φ(243) = 81 = Φ(486)/2
    φ(81) = 27 = Φ(162)/2
    φ(27) = 9 = Φ(54)/2
    φ(9) = 3 = Φ(18)/2
    φ(3) = 1 = Φ(6)/2
    φ(1) = 0

    There’s an analogy with part of this note on Heegner numbers:
    =
    The Heegner numbers greater than 3 can also be found using the Kronecker symbol, as follows: A number k [greater than] 3 is a Heeger number if and only if s = Sum_{j = 1..k} j * (j|k) is prime, which happens to be negative, where (x|y) is the Kronecker symbol. Also note for these results s = -k. But if s = -k is used as the selection condition (instead of primality), then the cubes of {7, 11, 19, 43, 67, 163} are also selected, followed by these same numbers to 9th power (and presumably followed by the 27th or 81st power). – Richard R. Forberg, Jul 18 2016
    =
    https://oeis.org/A003173

    The backbone is powers of 3. Complimentary stems are thus half or double opposing zipper teeth (or 3/2 or 2/3 looking the other way).

    What I’m outlining here is equivalent to what’s illuminated in MaKay & He (2015).

    Above I noted 27+1=28, 56+2=58, 240+3=243. 243 is on the zipper, anchored to the largest Heegner number 163 via 162. 58 & 28 are not on the zipper. Some readers may get confused because 54 & 27 are on the zipper.

    To avoid confusion: separate the two 27s conceptually by pathway.

    One is on the zipper and the other is derived from a peripheral structure that ties 163 to 59 through 104 (detailed in a few earlier comments). Go down the totient chain from 59 to 58 to 28 to get 58-2(bitangent)=56 and 28-1(line)=27.

    27=27 no doubt, but keep the pathways sorted when parameterizing topological course. Looking at the wrong stem leads to an incorrectly parameterized model.

    Go up the zipper from 162 to 243 to get 243-3(tritangent)=240. In the other direction that chain goes down through 54 & 27 — and that path should NOT be confused with the path through 59, 58, and 28 on another stem (a separate structure adjoined by a 104 bridge).

    Crystallized awareness of links and routes should (not trying to be naive about how 5 politics equals 2+2 in hard D-Orwellian times) eliminate the risk of misunderstanding.

    String.in the links back so far covered in he’er:
    perfect totient
    59 stem
    162 stem — tied up with top Heegner 163 & ally caught 4 further perfection s(652) lead.in he’er ET IC to 496
    ● Most won’t be ready for top-level Heegner perspective …with type O’s left.in to folly tempt UN fact-chuckers dare UN 2 fix major typos USing later “notation alert” (find.in maybe stew arts O-port-tune-IT 4 OBsurveysUN….)

    The term “totient” was coined by Sylvester. How fitting that his greedy double-exponential sequence broke the perfect solar system superstring symmetry with period doubling symmetry (more on U-N in the Jovian reference frame soon….including a 0% error).

  72. Paul Vaughan says:

    D-visor Some Luck D-own Less UN

    Things you might never stop to realize:
    √Δ(√(73500/15)) = √Δ(70) = √25 = 5
    σ(σ(σ(σ(Δ(√(73500/15)))))) = 104
    σ(σ(σ(σ(σ(Δ(√(73500/15))))))) = 210 = 7#
    σ(σ(σ(σ(σ(σ(Δ(√(73500/15)))))))) = 24^2 = 576
    σ(σ(σ(104/2))) = 260 = σ(171)
    σ(104/4) = 42
    σ(260/4) = 84 = σ(44)
    σ(66) = σ(70) = 12^2 = 144
    σ(σ(836/4)) = σ(240) = 744
    σ(4270/2/5) = σ(s(4370)/2/5) = 496
    Weather UNeven nor not even Joe Biden nos. 2+2 climate politics are about √Δ(√(73500/15)).

  73. Paul Vaughan says:

    MacK Sum U-N PRai$e^(Year Comp. Aim)

    1/11/500/2
    11500
    111.5
    11/2 = 5.5
    1 / 11500 ~= 111.5(U-N)-5.5(J+S)

    Thus:
    171.406220509617 = 111.5/(5.5*(J+S)+1/11500)
    171.406220601552 = 1/(U-N)
    -0.000000053635 = % error

    1 / 489426 = 1/2/3/7/43/271 ~= -300J+904S-301U-301N
    Recall 489426 =2*3*7*43*271
    1204 = 2*(301+301) = 2*2*7*43
    = 1806-301-301 = 2*3*7*43 – 2*7*43 = 300+904

    29.3625733662892 = 1/((J+S-1/2/3/7/43/271)/1204+S)
    29.3625733662893 = 4/(J+S+U+N) = jovian harmonic mean
    0.000000000000 = % error

    Sylvester seek wins!

    55.6462717641964 = 1/(4*((J+S-1/2/3/7/43/271)/1204+S)-J-S)
    55.6462717641972 = 1/(U-N)
    -0.000000000001 = % error

    84.0168459111161 = 2/(4*((J+S-1/2/3/7/43/271)/1204+S)-J-S+(5.5*(J+S)+1/11500)/111.5)
    84.016845922161 = 1/U
    -0.000000013146 = % error

    164.791315682562 = 2/(4*((J+S-1/2/3/7/43/271)/1204+S)-J-S-(5.5*(J+S)+1/11500)/111.5)
    164.791315640078 = 1/N
    0.000000025781 = % error

    With complements, join $11 each turn.in a round at:
    271 = 1000-729

  74. Paul Vaughan says:

    59 and 1728 j-invariant

    59’s at the core of more than i$ geneR(ally known) to war e ^ Or[well]s in Mayan Victory.
    Simple beginnings: at r=1/3, 59’s the level of the big spike.

    M rep’s anchored productively ± 12:
    196883=47*59*71=(59-12)*(59)*(59+12)=59*(59^2-12^2)
    12 = Φ(Φ(Φ(59))) = φ(φ(φ(Φ(59)))) and thus:
    196883=59*(59^2-Φ(Φ(Φ(59)))^2)=59*(59^2-φ(φ(φ(Φ(59))))^2)

    This one is fantastic.
    j-invariant hookup swings simply ± 5:

    27=1728/(59+5)
    32=1728/(59-5)
    59=27+32

    54=2*1728/(59+5) —— recall 54 points directly to 47 & 71 at r=1/5
    64=2*1728/(59-5)
    59=(54+64)/2

    1728=27*64
    1728=32*54

    48=AVERAGE(32,64)
    36=HARMEAN(27,54)
    1728=AVERAGE(32,64)*HARMEAN(27,54)

    Compassionate Lefties: Do you imagine 1728=AVERAGE(32,64)*HARMEAN(27,54) as a response if you probe thoughts on climate? Sensible private thoughts are thus silent: “don’t speak” no. doubt. GCD(left,right)=1 signals no common factor in climate dialogue. It doesn’t mean you have to viciously hate and savagely push terrorizing financial lockdown into homelessness.

    Φ(59) = 58
    Δ(58) = 27
    s(58) = 32
    59 = Δ(Φ(59)) + s(Φ(59))

    No. brainer under present circumstances: The New Zealand National Party should NOT have campaigned to lower taxes. A prequisite is a generous campaign for superior math education — a noble imperative for civilizations facing challenges which can not be solved with politics.

    George Polya’s advice: “Solve a simpler problem.” Done:

    1728 = 2*Δ(Φ(59))*s(Φ(59))

  75. Paul Vaughan says:

    O-Bay AI lie CAUGHT

    Aliquot.in moonlight sights tree-top baby monde stirrin’ minimal faithful representation.

    Hunted by U-No. they’re predatorial owl, the cute test right fly yen squirrel $scrambled further up the tree…

    4270=2*2135=s(4370) ——– 4370 is B rep
    s(2135) = 841

    Knew wall along far left “insight” looked extreme mist (the major west stern fault). Count O[r]link.in project seas comm. pound.in e^(XpeRt) curry US IT “nos. B(est)” what-the-ignorant missed: UN dare “Bei. area” guide D-ants.

    841=836+5 —— another member of the +5 pattern (review)
    s(836+5) = 5# ——– recall that 836 is the smallest untouchable weird number

    s(841) = 30
    s(30) = 42
    s(42) = 54
    s(54) = 66 — seq.U-Ns continues as “ayaM” conΦDOn’t EU AI red e^no.?

    Doubly greedy Sylvester’s (double exponential) Sequence precisely marks period doubling symmetry from perfect jovian superstring symmetry. The subtle difference is several orders of magnitude smaller than the dominant marks of B, M, & Leech and thus it motivates careful contemplation of model detail diagnostics.

    B, M, & Leech in solar system order are crystal clear. On this trail an investigator is tripping over strictly equal fits that keep piling up (thousands of noteworthy things). Someone will discover a terse way to encode a monstrous array of fits …or if you prefer cynicism, that’s already a trade secret and there are difficult people prepared to go to dangerous lengths to keep a precious gem hidden.

  76. Paul Vaughan says:

    West Turn Security Makes 298 BIG Investments in Better Math Education

    Homeland innocents learned
    to crave unilateral freedom simply because
    on CR Ave., U-N ill at Orwell free doom be cause.

    Multilateralism — a word with rapidly increasing negative connotations — means bad dictators “representing” several billion savagely bully millions of good folks in consequently faltering democracies, just like they do to their “own”.

    In 1984 UN e^(thick hale) nos. antonym 4:2020 hindsight.

    Φ(11) = 10 = Φ(22)
    Φ(19) = 18
    Φ(23) = 22
    s(323) = 37 = Φ(149)/4 = Φ(298)/4 = s(298+5^2)
    s(152) = Φ(149) = Φ(298) = 148 = 4*37 O No!

    Emerge.in-C-meet-UN secure ET count sol: western citizens might use math to explore elements sov. climb ET ….if they no. ET = Euler’s Totient.

    d(2,1/2,10) = d(2,1/2,13) = d(2,1/2,22) = d(2,1/2,18) = d(2,1/2,37) = d(4,1/2,58) =
    d(2,1/2,Φ(22)) = d(2,1/2,Φ(11)) = d(2,1/2,104/8) = d(2,1/2,Φ(23)) = d(2,1/2,Φ(19)) =
    d(2,1/2,s(323)) = d(1,1/2,Φ(59)) = d(2,1/2,Φ(149)/4) = d(2,1/2,Φ(298)/4) = 104

    s(652)=s(4*163)=496=s(496) is perfect
    163-104=59 trade perfect seek routes indeed!
    Φ(Φ(59)) = Φ(58) = Φ(29) = 28 = s(28)
    Just divide 4 UNother puzzle piece:

    R(1,1/2,7) = 0.0679047747389632 = ⌊(e^√7π)^(1/1)⌉^1 – e^√7π = 4072 – 4071.93209522526
    298 = 4370 – ⌊e^√7π⌉ = 4370 – 4072 = 2*5*19*23-ROUND(EXP(7^(1/2)*PI()),0)

    With a maze UN guide dense IT’s AI a cure rat must fear. IC WHO invited sci11UNs bullies to demonstate the US mull tee laud orwell “leadership” “change”.

    If I were Donald Biden or Joe Trump I would NOT debate but simply state:

    1. Pandemic: I will not lock you down. Without delay we will stop lockdown freaks from causing you to go broke and become homeless. We don’t want you living in terror that they’re going to pull some kind of trick — which predators surely do, availed opportunity.

    2. Climate: Superior math learning outcomes in homeland math education are a security imperative. We dare not face the monstrous cost of keeping 4/5 of homeland citizens in the dark.

    1984 = 4*s(4*163) = 4*496 = 8/3 * 744 = 1728 + 104 + 152
    You really think George Orwell didn’t No.?
    “don’t speak” no. doubt

    70p is weird for all primes p ≥ 149

    1728=2*(836+28) ——– 28 is lowest perfect; 836 is lowest weird untouchable; 1728 of j-invariant
    1728=4*496-104-152=8*744/3-104-152=1984-104-152=8128-6400
    298=104+152+42=2/3*744-149-49=496-149-49
    =4*496-1728+42=8*744/3-1728+42=1984-1728+42 hitchhiker guides orwell too jovi-invariant

  77. Paul Vaughan says:

    D-anger
    Steep Cliffs
    Stay on the Trail
    Φ(85) = 64
    Φ(128) = 64
    Φ(136) = 64
    Φ(160) = 64
    Φ(170) = 64
    Φ(192) = 64
    Φ(204) = 64
    Φ(240) = 64 ~~~~~~~ “I’m goin’ 2 Whishh! IT awe” — The White Stripes
    ^v^v^v^v^v^v
    Φ(64) = 32
    Φ(32) = 16
    Φ(16) = 8
    Φ(8) = 4
    Φ(4) = 2
    Φ(2) = 1
    Φ(1) = 1
    240 = 1728 mod 744
    “make the sweat drip out of every pore”
    271 = 1729 mod 729
    Φ(241) = 240
    Φ(287) = 240
    Φ(305) = 240
    Φ(325) = 240
    Φ(369) = 240
    Φ(385) = 240
    Φ(429) = 240
    Φ(465) = 240
    Φ(482) = 240
    Φ(488) = 240
    Φ(495) = 240
    Φ(496) = 240 ~~~~~~~~~~~~ “a 7 ACE shh!U-N…” – TWS
    Φ(525) = 240
    Φ(572) = 240
    Φ(574) = 240
    Φ(610) = 240
    Φ(616) = 240
    Φ(620) = 240
    Φ(650) = 240
    Φ(700) = 240
    Φ(732) = 240
    Φ(738) = 240
    Φ(744) = 240 ~~~~~~~~~~~~~ “cou dn’s top…” ———————–
    Φ(770) = 240
    Φ(792) = 240
    Φ(858) = 240
    Φ(900) = 240
    Φ(924) = 240
    Φ(930) = 240
    Φ(990) = 240
    Φ(1050) = 240
    ^v^v^v^v^v^v^v
    Φ(240) = 64
    Φ(64) = 32
    Φ(32) = 16
    Φ(16) = 8
    Φ(8) = 4
    Φ(4) = 2
    Φ(2) = 1
    Φ(1) = 1
    “…and dem. B[]den right be 4 the lord” — Jack Black

  78. Paul Vaughan says:

    Tour Key’s Perfect Neat O-String.in the French Moonlight

    496 = 23*19 + 59 ———– If you don’t know 496 is perfect by now, I’ve lost hope in EU….
    496 = 4370/2/5 + 59 ——- …and dialogue is finished as I know EU stand WITH ORWELL.

    MLB is major league baseball — and monster leech baby if math UN no. lingo.

    496-69=s(4370)/5/2=4270/5/2
    225=d(4,1/5,54)=d(3,1/2,9)=(73500/70^2)^2=298-73=496-271
    24=(15^2-71-59-47)/2 ——————- 15 The O REM anyon?
    162=-15+71+59+47=104+Φ(59) —— perfectly totient orient a s√(Φ-φ)un
    163=104+59 ——————————- Heegner tip love a11 prime

    Rudolph: With well-muzzled math hawk key stick to 6.

  79. Paul Vaughan says:

    West Turn President’s Hexagonal Rev You

    Innocent shorthand for hexagonal is deliberately misinterpreted within Macron’s moonshot on the deep state USS C[e^NSO]Rship.

    Rightly We left IT a bit of mist eerie as Macron’s busy trick king Boris’ into luck D-own financial terror, “leading” to homelessness in the “home” land.

    Quick Rev EU: Hexagonal shorthand isn’t PC in the “home” land so (“don’t speak” no. doubt) weave just numbers:

    7 = 3*(1*2)+1 ——– thank God R(p,1/2,7) separates fact-chuckers from their miss UN of “truth”
    19 = 3*(2*3)+1 —— = 28-9 seen in UN SIM MET try eclipsed by Sylvester’s Sequence
    37 = 3*(3*4)+1 —— = 28+9 ditto; s(196883-196560)=37; d(2,1/2,s(196883-196560))=104
    61 = 3*(4*5)+1 —— JS
    91 = 3*(5*6)+1 —— ties to 7 & 104 through 13 (the[a]me√(Φ-φ).in refer UNs frame)
    127 = 3*(6*7)+1 —– CO2shh!UN = 5 ^ 2 Be/11 O
    169 = 3*(7*8)+1 —– square root = 13 fits 104
    217 = 3*(8*9)+1 —– ties 7 to 31 (see next line)
    271 = 3*(9*10)+1 —- 1729 mod 729 = 271 & 1728 mod 744 = 240; difference = 31

    Why’s party shh!UN.in sight?

    127 * 2 = 254 = 323-69 = 196883 – 196560 – 69
    298 = 254+44 = 323-25
    where totient Φ(n) + cotient φ(n) = n weather off or on climb ET (Euler’s Totient) :
    Φ(69) = 44
    φ(69) = 25 = 5 ^ 2

    Politicians like Macron are eager to misinterpret, misunderstand, and misrepresent (mmm) the numbers.

    IT’s a fly ball to left field shall O UN dr. the “green” monster in F(UN) weigh:
    69 = Φ(69) + φ(69) = 44 + 5 ^ 2
    Pen Syl v[a]n “O ya!” Fall O red “duh!”: Clyde Bollinger ’52 “hawks”agon reversed KO peek. in tour est 44.

    Macron’s west turn so-called “PC” Orwellian “Hi Jack!” is a serious, mature subject.

    397 = 3*(11*12)+1 = 496-9*11 = 298+9*11 = (196883-196560) + 2*s(196883-196560)

    We can peacefully stop IT by whatever just means as necessary.

    We started on this trail a few months ago. SUDDENLY it became necessary to SHARPLY alter course. Why? Rightly left a total mystery (totient if you prefer).

  80. Paul Vaughan says:

    CO2shh!UN + Möb√(Φ-φ)US

    δ(58825) = 15625 = 5^6
    δ(15625) = 3125 = δ(3990)
    δ(3125) = 625
    δ(625) = 125 = δ(186)
    δ(125) = 25 = δ(46)
    δ(25) = 5 = δ(6) = 5^1
    δ(5) = 0

    With weather mist right.in Eur. face msm fact-chuckers left with no. clue:

    4 = δ(8)
    3 = δ(9)
    7 = δ(10)
    0 = δ(11)

    “[e].g.ive a11[i.e.] seek R-IT’s[CRackCuRe^s]uawei” — [Tie]WonR-PubLock-D[oorwell]

  81. Paul Vaughan says:

    F(UN)D-red$hawk$park$way over Eur. rise mod e ln blue moon shine shh! ale O-in left [F]ool [w]ields.

    Ware ET = Euler’s Totient inXS “[]ive got tool ET UNo.” link.in PR Joe Act found door w[oo]l.

    σ(120) = 360
    σ(174) = 360
    σ(184) = 360
    σ(190) = 360
    σ(267) = 360
    σ(295) = 360
    σ(319) = 360
    σ(323) = 360 = Φ(427)=Φ(7)*Φ(61)=6*60 = Φ(836) = φ(504) ~~~~~~~~ s(n)+n=σ(n)
    σ(359) = 360

    Seas Norton’s “voice of God” past link.in “PR Joe Act”:
    s((496-59)*5*2)/5/2-323=104=496-Φ(69)-φ(69)-323=496-σ(Δ(69))-Δ(70)-323
    104=d(2,1/2,s(323)=d(2,1/2,Φ(59))=d(4,1/2,Φ(59))
    323=196883-196560=360-s(196883-196560)=298+25=2*Φ(163)-1
    Like Bernie remember 28 & 496 are perfect in nature.

    in baby steps:
    s(323)=1+17+19=37
    σ(323)=1+17+19+323=360

    With semiPRhyme e-tees math theme ET IC shh! UN’s in monster US red white & bull EU moonlight blew:
    323=17*19
    437=23*19=496-59
    4370=23*19*5*2
    4270=s(4370) =70*61=61*(1^2+2^2+3^2+…+22^2+23^2+24^2)
    427=s(4370)/5/2=7*61=496-69=496-44-25=323+104

    “O-Bei!” WHO? oooo Shh! IT O[]well superstring bye D-UN catch a Be/11/e “soft power”.
    “Luck D-own!!” Witch Ale Jack O-Land Turn Dr.ink$ at UN?

    Hexagonal right and left 2 get there in “Bernie’s Perfect!!” bury center SIM ET tree:
    s(496-Φ(69)-φ(69))=s(496-69)=69=Φ(69)+φ(69)=(7+37)+(1+61-37)=1+7+61=s(427)=s(7*61)

    Ice break snow ghost 2 sea past climb ET:
    28=(19+37)/2

    Sox Red O canal, B(est) to knock. WHO’s heir? No. weigh! Dare in.knacks!hhhex ” ’cause’: I’m knots leap.UN “ with “green” monde stir f(right).in left Fields Rush Macron nom IC mmm ET a11: “I will choose free w[oo]l”.

  82. Paul Vaughan says:

    Another important one:
    σ(427) = 496

    Semiprime 427 is the product of hexagonal numbers 7 & 61 and a factor of s(4370)=4270.

    The way Baby Monster, Monster, and Leech representation weaves with perfect and hexagonal numbers is both simple and spectacular.

    Begin aside.
    Probing UN scaling to next-order (from first-order 4270) turns out to be quick, easy exploration:
    1.00002190780229 = harmean(1.00002638193018,1.00001743371442)
    *70*61 =
    4270.09354631576
    4270.09258127429 — UN slip cycle Seidelmann (1992)
    0.000022600013 = % error
    End technical aside.

    Begin more general commentary.
    The curious thing I note as a cross-disciplinary hybrid cruising through these readily pliable waters is the absence of linkages between works addressing A & B:

    A. totient, aliquot, divisor sum, mobius, cototient, and related sequences.
    B. monstrous moonshine.

    The void is spectacular, as the linkages look like trivially definitive foundations.

  83. Paul Vaughan says:

    Nos. Reign D-air Left as a Mist Eerie

    Tie won: solve est tour seek wins “D-own in history” of “free” dim.
    s(437) = 43 = Δ(69) = Δ(s(427))

    “no. Eur.a11y g(ET) me g(O-win) when EU Put. IT a11 ln….”[J]anko [D]ones

  84. Paul Vaughan says:

    Hale O-We’UN Climb ET Zipped Past Sci11UNtest

    Perfect PET flyin’ squirrel glides escape UN “wise” sold “owl”.

    =
    One of the many continued fraction expressions for e is 2+2/(2+3/(3+4/(4+5/(5+6/(6+ … from Ramanujan (1887-1920). – Robert G. Wilson v, Jul 16 2012

    e maximizes the value of x^(c/x) for any real positive constant c, and minimizes for it for a negative constant, on the range x [greater than] 0. This explains why elements of A000792 are composed primarily of factors of 3, and where needed, some factors of 2. These are the two primes closest to e. – Richard R. Forberg, Oct 19 2014
    =
    https://oeis.org/A001113

    Why sold o[r]w[el]l woolover rise in 1984?

    =
    a(n) for n [greater than or equal to] 1 is a paradigm shift sequence with procedural length p = 0, in the sense of A193455. – Jonathan T. Rowell, Jul 26 2011
    […]
    =
    https://oeis.org/A000792

    A193455 dials EU.in the loop: “Paradigm shift sequences: A000792 (p=0), […]”

    Tool of Orwell in the bet PR pro g,h, and i names will sun for borg rOw(e/1)1 to board of fly yens$quarrel climb ET direct tears (“don’t speak” no. doubt).

    Adjacent totient pairs difference and cototient pairs sum to the 3rd set of zipper teeth:

    324 = 486-162 = 243+81 = 2*162 = 196883-196560 + 1
    108 = 162-54 = 81+27 = 2*54
    36 = 54-18 = 27+9 = 2*18
    12 = 18-6 = 9+3 = 2*6
    4 = 6-2 = 3+1 = 2*2

    Orient all PET sense soar well, fur “green” monster US climb ET.

    “Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on […….]” — yet another page in the sci11UNs moonlight.

  85. Paul Vaughan says:

    30031 & 509 Review with Sylvester Sequence

    Recent insights solved an old puzzle.

    30030 = 13*11*7*5*3*2 ——— primorial 13#
    30031 = 13*11*7*5*3*2+1 = 59*509 ——- 30031 special property: first primorial+1 that’s not prime

    510510 = 17*13*11*7*5*3*2 ——— primorial 17#
    1001 = 13*11*7
    510 = 17*13*11*7*5*3*2 / 13*11*7 = 17*5*3*2
    509 = 30*(17) – 1 ——– where primorial 5# = 30
    30031 = 30*(13*11*7) + 1
    59 = (30030+1) / (510-1) = 30031 / 509
    509 = 30031 / 59

    30031 bridges through moonshine to Sylvester’s Sequence.

    1806=2*3*7*43
    301=1806/6=7*43=509-208
    208=509-301
    104=(509-301)/2=(509+323)/8

    Placeholder 71000 painted a simple first-order modular demo above to prelude next-order sharpening. To within a module the same number turns up here tying JS to UN with a multiple of 509.

    This stresses that double exponential greed settles the details of the last internal partition.

    We see J, S, & U in accord with what we understand of B, M, & Leech and then as for why N (the last giant) is off ever so slightly from comparative theoretical symmetries in a very precise manner, we now understand that settlement as a simple consequence of double exponential greed.

    The game plan here plus or minus 5 from 59 mean.in J[S]UN very ant 1728 = (59-5)*(59+5)/2. See cure roots PET no. green monde stirs in the paris owl on Eur. sense soar shh!!

    B partitions 42 to define U:

    4270=s(4370)
    4370=2*3*19*23

    42=19+23=298-256

    298=1984-1728+19+23=323-25
    256=1984-1728=104+152

    whishh! IT awe:

    298=279+19
    256=279-23

    85=279-194 —– see MacKay & He’s (2015) “Sporadic and Exceptional”

    84.0168459218229 = 194/(-85/11.8626151546089+279/29.4474984673838-1/36750.3379015986-1/30031)
    84.016845922161 = 1 / U
    -0.000000000402 = % error

    36750, 30030, or both together give a slightly larger but still vanishingly small error.
    Double-acts pawn. Ants yell: “solve est dare.” Seek wins taxicab dis(put)in counterbalUNs.

    1 / 489425.981694385 = -300J+904S-301U-301N
    1 / 489426 ~= -300J+904S-301U-301N
    489426 = 2*3*7*43 * 271

    1 / 489426 +300J-904S+301U ~= -301N
    1/N ~= -301/(1/2/3/7/43/271+300J-904S+301U)
    164.791315640071 = -301/(1/2/3/7/43/271+300J-904S+301U)
    164.791315640078 = 1/N
    -0.000000000004 = % error

    Laskar converges to 36750.3379015986 not 36750 = 73500/2.
    30031 primorial+1 has special property as noted above.

    Suddenly 4*496 superstrings 1728+298-42 no. mystery in 1984. Left IT shelved right up UNtil note far above point-D to 4270 & 1584 from 4370.

    Last spring the missing link was math taxonomy keyed nowhere near biology cataloguing standards to “solve a simpler problem” left right hearin’ “home” land D-vision.

  86. Paul Vaughan says:

    US: Retune IT Deep O-Locked to Jo.vi.UN.axis

    Dozens of comments I had prepared months ago were deleted following an abrupt change of course (~2 months ago). An excerpt of 1 was delayed rather than deleted.

    30.4320075307947 = (835.546575435631)*(29.3625733662892) / (835.546575435631 – 29.3625733662892)
    65.0170708690834 = slip(30.4320075307947,19.8650360864628)
    708.556226479596 = slip(65.0170708690834,19.8650360864628/4)

    This relates back to OB’s comments on the 2400 thread.

    708 = 4*(47+59+71) = 12*59

    […] speak openly about catering to the support of those “fed up” by state restrictions […] encouraged chants […] calling for the imprisonment of local officials who have instituted them.”

    Exceedingly few jurisdictions have sensible, well-balanced pandemic leadership.

    For abusive jurisdictions where extreme measures are taken (things we expect only in places under the control of The Communist Party) I agree in resolute solidarity with calls to swiftly and decisively remove from public office those savagely pushing psychological and financial terrorism against their own people.

    It should be ensured that they — and all others in our society — are able to remain financially stable and adequately housed. Euthanasia should be made readily available hassle-free to those without the will to deal with strictly unbearable tyrants who are — for whatever contrived reason — to remain in power.

    Lockdowns never occur in free countries, so a mark of liberation leading nations can pursue is swift alteration of their constitution.

    I’m repeating the following to underscore that simple groups are tied perfectly to hexagonal numbers:

    28=(19+37)/2

    s(496-Φ(69)-φ(69))=s(496-69)=69=Φ(69)+φ(69)=(7+37)+(1+61-37)=1+7+61
    =s(427)=s(7*61)=496-7*61=σ(427)-427=496-323-104=496-(196883-196560)-104
    =111-42 where 111 is Lyons rep (see first 2 embedded links in next comment)

    So-called “experts” could have simply pointed this out a dozen years ago.
    Weather ignorance or deception: dark either way — and undermined trust.

  87. Paul Vaughan says:

    Jan. KO Lie UN’s Group Representation

    104=2*(8*111-836)=d(2,1/2,37)=d(2,1/2,s(196883-196560))
    744=d(3,1/2,19)=d(3,1/2,43)=d(3,1/2,67)
    28056 = d(3,1/2,37)
    28=MOD(d(3,1/2,37),1333-3*111)/2=28=ROUND(d(3,1/2,37)/(1333-3*111),0)=s(28)
    “what I saw that night was real and not just fantasy” — ire UN made dem
    7920=5256+(836+496)+(1333-1)
    7920=5256+24*111
    1333=43*(MOD(1729,729)-MOD(1728,744))=43*31
    271=MOD(1729,729)=1333-3*111-3^3^2=s(4370)-3*1333=4270-3*1333
    240=MOD(1728,744)
    163=496-3*111
    φ=-2*SIN(2π*(σ(427)+836)/(Φ(836)+Φ(427)))

    “monster group”? “baby monster”? What names could more effectively discourage and undermine public discussion of “simple sporadic groups”? Perhaps only “number of the beast” (suspiciously located exactly at the cross-roads of perfect superstring, smallest weird untouchable, Janko & Lyons).

    The nomenclature chosen by mathematicians sure doesn’t make for a very palatable mainstream message.

    I’ve become quite cynically suspicious that these names were deliberately chosen to “help” make public mention appear toxicly extreme mist. “don’t speak” no. doubt kind of thing to encourage just knowing about it privately for share ring with an “insider” group.

    Whatever stupid or just plain inconvenient names the mathematicians want to call these things, they all fit together perfectly in a puzzle that isn’t even difficult.

    Of course the crazy names do very strongly aid memory, as does the song “monsters in the paris sol” by queens of the stone age ….especially since 4 times perfect superstring number 496 = 1984. As they say in the song: “O[]well”.

    We might as well decide even if only just for f(UN) that weave proof of a biased ant math group dismissing simple logic on the basis of attached names. Imagine a “Donald Trump Theorem”: “O well” then no. answer needed by dem work-around.

  88. Paul Vaughan says:

    The Cross Roads Sov. Representation Theory

    111=3*37
    111=3*s(196883-196560)

    Luck D-own: WHO’s at UN?
    (836+496)/2=(1333-1)/2=2*3*111

    Someone had fun naming all these things — satisfied their supervisor IC.

  89. Paul Vaughan says:

    5256 = Φ(111)*φ(1333)
    37=111/3=δ(1333)/2=s(196883-196560)
    104=d(2,1/2,37)=d(2,1/2,111/3)=d(2,1/2,δ(1333)/2)=d(2,1/2,s(196883-196560))
    163=104+59=111+52=2*111-59=47+71+104-59
    111=(47+71+104)/2=59+104/2
    1333=43*(2*71-111)
    Δ(111) = 71

  90. Paul Vaughan says:

    “Start Again”

    “There is no. thing fair in this world.”
    836 = 1333-497
    497 = 1333-836

    “There is no. thing safe in this world.”
    2432 = 2*1216 = 4*608

    “….and there’s no. thing sure in this world.”
    496 = s(496) = s(652) = s(s(608))
    8128 = s(8128)

    1000 = s(s(1216)) = 1333 – 3*111
    640 = φ(1216) = δ(1216)
    320 = φ(608) = δ(608)
    “…and there’s no. thing pure in this world:”

    640320 = s(s(1216))*φ(1216)+φ(608) = 1000*640+320
    640320 = s(s(1216))*δ(1216)+δ(608)
    640320 = 69*(1152+8128)

    1152 = δ(1728) = φ(1728) = Φ(2432) = Δ(2432) = 2*σ((3*5*7*11*13)/(3+5+7+11+13))
    576 = Φ(1728) = Δ(1728) = Δ(1216) = Φ(1216) = 24^2 = σ(210) = σ(497) = σ(1333-836)

    576 = σ(385) = σ((3*5*7*11*13)/(3+5+7+11+13))
    385 = (3*5*7*11*13) / (3+5+7+11+13)

    “Look for sum thing left.in this world. ” — Billy IDo[rwel]l

  91. Paul Vaughan says:

    Green Monster “Free” Will eXpeRt Orw[oo]lover Eur. Rise

    “Hey little sister, WHO is IT Eur. with?” — Billy I[woo]l

    δ(2*Δ(Φ(59))*s(Φ(59))) = φ(2*Δ(Φ(59))*s(Φ(59))) = 8*(59^2-196883/59)
    Φ(2*Δ(Φ(59))*s(Φ(59))) = Δ(2*Δ(Φ(59))*s(Φ(59))) = 4*(59^2-196883/59)

    2*Δ(Φ(59))*s(Φ(59)) = 1728
    2*δ(Φ(59))*s(Φ(59)) = 1984

  92. Paul Vaughan says:

    Link 2

    Φ() = euler’s totient
    https://oeis.org/A000010
    https://oeis.org/A000010/b000010.txt

    φ() = cototient
    http://oeis.org/A051953
    http://oeis.org/A051953/b051953.txt

    notation overlap distinction:
    golden ratio if no (argument’s in brackets)

  93. Paul Vaughan says:

    Link 3

    Δ(n) = Φ(n) – μ(n) = totient – möbius
    http://oeis.org/A053139
    http://oeis.org/A053139/b053139.txt

    δ(n) = φ(n) + μ(n) = cototient + möbius
    http://oeis.org/A228620
    http://oeis.org/A228620/b228620.txt

  94. Paul Vaughan says:

    Remember: 652 = 4*163

    This stuff is stunning. It’s such a pleasure to work on it and see all the connections.

  95. Paul Vaughan says:

    TheO[rwell]Airs:IDweight

    What curry US IT?
    2*s(s(Φ(59)))*s(Φ(59)) = 1984 = 4*496 = 8/3*744
    D:phi calltosea witchpressOdauntiewon ?
    2*Δ(Φ(59))*s(Φ(59)) = 1728 = 1984-104-152 = 1984-298+42
    e con nom IC pope EU lost: luck “D-own!!” with Joe win sob miss UN to comm.UN.us-part e:
    2*δ(Φ(59))*s(Φ(59)) = 1984

    “I can still recall the time: wait!..TRIED to breakONthrew to THEO[]airsID” — The DOORS

  96. Paul Vaughan says:

    Press O “Don’t!”

    Another “perfect” name they came up with was “pariah”.

    496 = s ( 652 ) = s( s ( 608 ) ) = s(496)
    496 = s(4*163) = s(s(2432/4)) = s(496)

    I’m convinced “elite” “leaders” plan to give western “working class populists” the luck D-own D-ea[r]th bull O for not be[]ing quick fall O-in a “tie won” plan. Just.in took “aha!” stage hop.in IT’s just UNoff dawns a port forge[]O2 sea just in.history’s trait show door.

    “Hey Joe, ware EU go!in with that … .. …. …. ?” — j.handrocks

  97. Paul Vaughan says:

    The DOORSequence

    “the gate is strait
    deep and wide”

    1, 2, 3, 7, 11, 19, 43, 67, 163

    42 = 1*2*3*7 = 7# / 5

    “found.D-UN.island.in.Eur.arms
    count.try:in.Eur.rise
    arms that ch[a]in US
    e^yes that Lie” — the doors

    489426/7/3/2/1=43*271
    Φ(7#)/2=1+2+3+7+11=24
    43-24=19
    43+24=67

    Sylvester coined the term “totient”.

    163=271-108
    162 = Φ(163) = Φ(162) + φ(162) = 54 + 108 = φ(7#) = 7#-Φ(7#)

    (19+43+67+163)/2=146
    146-42=104
    163-104=59; 59-12=47; 59+12=71
    average(71mod24,59mod24,47mod24)=average(23,11,23)=19=43-24=67-48=67-Φ(7#)

    “EVorrery body loves my baby”

    the number 24 is the only integer bigger than 1 with this property

    67+43-19=91 shh!..

  98. Paul Vaughan says:

    1806 is no longer a mystery. I introduced the theme on the last XR thread. One of the most important comments I’ve ever submitted never appeared. That led to a decision to sharply alter course, similar to when I began my strict boycott of WUWT & CE about 4 years ago.

    As with the solar terrestrial weave (cyclic volatility of semi-annual mid-latitude westerly circulation at Schwabe-timescale), 1806 is conclusive. Monstrous moonshine ties Farey sequences and Ford circles through Egyptian fractions to Heegner numbers. Sylvester’s sequence bounds solar system order. (The final puzzle piece was the 2-page mathematical proof — somewhere down the link-trail I gave over here.)

    =
    Thus we are asking for integers which make the expression
    1 − 1/p − 1/q − 1/r
    strictly positive and as small as possible. This minimal value is 1/42, and
    1 − 1/2 − 1/3 − 1/7 = 1/42
    =

    The first 4 Heegner numbers:
    1, 2, 3, 7 — and their reciprocals:
    1, 1/2, 1/3, 1/7

    Explore some properties:
    1/(1-1/2-1/3-1/7)
    1*2*3*7

    Note well:
    1+2+3+7+11
    average(19,43,67,163)
    73 mod 24
    So the Heegner numbers point directly to the lowest prime congruent to 1 mod 24.

    CT(ET(163))=2*ET(ET(163))=2*(11+43)
    271=163+CT(ET(163))
    The first 4 primary pseudoprimes are constantly offset from Sylvester’s sequence.
    (CT=cototient; ET=Euler’s totient)

    Once you start looking at how all of the pieces fit together it becomes really simple.
    There are plenty more examples beyond these:

    52=323-271=323-163-108

    496=271+2*177-129
    496=323-52+2*177-129
    496=196883-196560-52+2*(47+59+71)-(19+43+67)
    496=271+129+96

    177=(47+59+71)
    129=(19+43+67)

    271=323-52=163+108=298-27
    323=271+52=196883-196560
    196883=(59-1728^(1/3))*(59)*(59+1728^(1/3))=47*59*71=59*(59^2-1728^(2/3))
    884736=2*(1984-1728)*1728 (That’s a sample from a stack that IDs everything from solar equatorial rotation through Chandler, QBO, perigy, evection, and whatever else you want to name.)

    194=323-129
    323=194+129
    129=323-194
    48=177-129

    225=2*(47+59+71)-(19+43+67)=2*177-129=354-129
    225=48+(47+59+71)
    225=96+(19+43+67)

    48=225-177=225-(47+59+71)
    96=225-129=225-(19+43+67)

    62=225-163
    208=177+31
    194=225-31
    225=163+43+19
    31=(225-163)/2=208-177=225-194 = average(19,43,67,163) mod 42 = 73 mod 42
    42=1/(1-1/2-1/3-1/7)=1*2*3*7
    298=(1984-1728)+42=(152+104)+42=225+73
    s(298)=225-73=152

  99. Paul Vaughan says:

    Whatever’s Mayan

    “The last free daze:
    The reign was UN. Stop a bull.” — Tom Petty

    1 = 1 mod 24
    2 = 2 mod 24
    3 = 3 mod 24
    7 = 7 mod 24
    11 = 11 mod 24

    The 5 lowest Heegner numbers sum to 24.

    19 = 19 mod 24
    19 = 43 mod 24
    19 = 67 mod 24
    19 = 163 mod 24

    The 4 highest Heegner numbers average 73.

    1 = 73 mod 24

    73 is:
    the first prime above top-level M prime 71.
    the smallest prime congruent to 1 mod 24.

    Remarkably none of the introductory materials I encountered on Heegner numbers pointed this out.

    “Workin’ on a mystery — goin’ wherever it leads
    I’m pickin’ up whatever’s Mayan” — Tom Petty

  100. Paul Vaughan says:

    Review

    Number of fractions in Farey series of order n

    Compare with: Sum of totient function
    “James J. Sylvester, On the number of fractions contained in any Farey series of which the limiting number is given […] (1883)”

    |F_1|=2
    |F_2|=3
    |F_4|=7
    |F_11|=43
    |F_29|=271

    1806=|F_1|*|F_2|*|F_4|*|F_11|=2*3*7*43
    489426=|F_1|*|F_2|*|F_4|*|F_11|*|F_29|=2*3*7*43*271

    Recall that we’ve discussed how this shows up precisely in Jovian order.

    A little more tricky to fathom, but should be noted:
    3=|F_2|
    29=|F_9|=|F_3^2|=|F_|F_2|^2|
    271=|F_29|=|F_|F_9||=|F_|F_3^2||=|F_|F_|F_2|^2||

  101. Paul Vaughan says:

    Orientation

    You can derive euler’s totient from farey.
    You can derive mobius from farey.

    Thus you can derive e from farey (recall Schneider’s classic).

    There’s a cute, simple trick for generating farey sequences using floor functions. It’s among the most delightful little piece of magic I’ve ever encountered in mathematics. See equation 3:
    =
    If a/b and c/d are neighbors in a Farey sequence of order N, the next term in the same sequence is given by the following expression:
    ⌊(N+b)/d⌋*c – a [numerator]
    ⌊(N+b)/d⌋*d – b [denominator]
    [equation] (3)
    where ⌊x⌋ is the floor function. Therefore, the Farey sequence of order N can be generated very efficiently starting from the elements a/b = 0/1 and c/d = 1/N.
    =

  102. Paul Vaughan says:

    Typo: “primary pseudoprimes” should read “primary pseudoperfect numbers”

  103. Paul Vaughan says:

    Gathering some links all in one place for convenience:

    1, 2, 3, 7, 11, 19, 43, 67, 163 — Heegner
    2, 3, 7, 43, 1807, … — Sylvester
    2, 6, 42, 1806, … — primary pseudoperfect
    1, 2, 6, 42, 1806 — intersection
    1806 — pinpoint

  104. Paul Vaughan says:

    Here’s a noteworthy sequence that’s not in the OEIS — sums are from 0 to N :

    Σ|F_0|=1
    Σ|F_1|=3
    Σ|F_2|=6
    Σ|F_3|=11
    Σ|F_4|=18
    Σ|F_5|=29
    Σ|F_6|=42
    Σ|F_7|=61
    Σ|F_8|=84

    A few other noteworthy values:
    Σ|F_10|=146
    Σ|F_14|=360
    Σ|F_15|=433

  105. Paul Vaughan says:

    The math affords us many avenues for equivalent expresssion.
    Oldbrew may view Sylvester’s 2, 3, 7, 43, 1807 Sequence in this light:

    2*2-1 = 3
    2*3-1 = 5
    2*7-1 = 13
    2*43-1 = 85
    2*1807-1 = 3613

    Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right triangle. 3, 5, 13, 85, 3613, 6526885,

  106. Paul Vaughan says:

    mods: right triangle spiral translation for OB is stuck in the filter

  107. Paul Vaughan says:

    IT Ski Rev EU: “Win September” 1984

    Joe thank cue: Constitution of 1806 left in file truth tour press “I don’t tell” [B]lock-D from 2020 piece search. West turn-(D Clock)-back 36 years.

  108. Paul Vaughan says:

    Monster US Moon Ch!nes Won and UN[all]lie Pure[D]fect L!nkin “too Heegner” PR Joe Act

    Let us order sum partition of supersingular PR rhymes:

    24=2+5+17 = 71 – 47 = 67 – 43 = 43 – 19
    47=3+13+31
    59=7+11+41
    71=19+23+29
    177=47+59+71 = 3*59

    378 = 24+47+59+71+177 = 78 + 111 + 189

    78=2+3+7+19+47
    111=5+13+11+23+59 = 189 – 78 = 3*37 = 3*s(196883-196560) ————– Lie UNs Group
    189=17+31+41+29+71

    Sporadic simple group primes include only 1 perfect 496=s(496) D-visor:
    31 = 496/16 = ( 19 + 43 ) / 2

    378=2+3+5+7+11+13+17+19+23+29 +31 +41+47+59+71=1+2+3+7+11+2*19+2*43+67+163
    316=2+3+5+7+11+13+17+19+23+29 -31 +41+47+59+71=1+2+3+7+11+1*19+1*43+67+163

    129=2+3+5+7+11+13+17+19+23+29 = 19 + 43 + 67 = 2*31 + 67 = 3*43
    201=2+3+5+7+11+13+17+19+23+29+31+41 = 129 + 3*24 = 3*67

    3^2 = 129 mod 24 = 201 mod 24 = 177 mod 24
    3 = 129/43 = 201/67 = 177/59 = 111/s(196883-196560)

    s(129) = 47
    s(201) = 71

    496=378+47+71=378+2*59
    D-liberate Am. BIG glue IT: Donair weather profoundly simple or(well)s imply PR(O)f(O)UN-D()Near-tune herd “the voice of God”.

  109. Paul Vaughan says:

    111=Φ(323)-(47+59+71)=Φ(196883-196560)-(47+59+71)
    Ly : 163=2+3+5+7+11+31+37+67

  110. Paul Vaughan says:

    “The 15 supersingular primes: primes dividing order of Monster simple group.”
    2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71

    Put them in the 3 by 5 block (as I did just above). Add rows. Add columns. Clue in.

    2 5 17
    3 13 31
    7 11 41
    19 23 29
    47 59 71

  111. Paul Vaughan says:

    Count Sol Clear Ants Or Well

    Luck D-own human writes “eXpeRt” WHO’s at UN nos. ware seek “cure IT” counts ill f(IT)sin.

    177 = 47+59+71 = 3*59

    y = (177-((19+43+67)-s(196883-196560))/2-59)^5 – (((19+43+67)-s(196883-196560))/2)^5
    x = 19^5 + 43^5 + 67^5

    37 = s(196883-196560)
    Monster US Jan. KO Lie UNs group Bei. B:
    47 = s(19+43+67) = (y-x)^(1/5) = s(3*s(4370/5/2))
    71 = s(67+67+67) = s(s(Φ(163)))

    “ON my blew PR INT (IT SIM f**** O!N) UK” — LG

  112. Paul Vaughan says:

    breakfast ware the news is red — the doors

    177=288-111=47+59+71=288-(189-78)
    Φ(323) = 288

    496=378+177-59
    496=378+288-111-59
    496=378+Φ(323)-111-59
    496=378+Φ(Φ(1333))-111-59
    496=378+Φ(196883-196560)-111-59

  113. Paul Vaughan says:

    Sea Lie Way?
    BRI be. $0[]well!

    Sporadic simple “pariah groups” J4 & Ly bridge perfectly (i.e. s(496)=496) in monster US moon shh! UN from “too super” sing allure “tee hee” he ignores in afar age review of factorized order call ’em.

  114. Paul Vaughan says:

    6, 28, & 496 are Perfect

    Note how they piece together with the top Heegner (163) prime (wit chisalls O some of Lie UN PR rhymes), the top 3 monster group (M) primes (47, 59, 71), and the “minimal faithful representation” of Ly & J4 (111 & 1333).

    s(608) = 652 = s(4*152) = s(2432/4)
    s(652) = 496 = s(4*163)
    s(496) = 496

    1260=608+652=Φ(1333)
    1260=2432/4+4*163
    1260=4*152+4*163
    152=118+28+6=2*59+28+6=(47+71)+28+6=(177-59)+28+6=(288-111-59)+28+6
    =(Φ(323)-111-59)+28+6=(Φ(1260)-111-59)+28+6=(Φ(Φ(1333))-111-59)+28+6

    Alternate aspects partially overlap puzzle-explore “Or” awareness US eerie to bridge mainstream gaps.
    Again: The role of perfect numbers in ordering sporadic groups simply appears, overlook-D$buy$Convention.

    1333=1260+73
    1333=652+608+73
    1333=4*(163+(Φ(Φ(1333))-111-59)+28+6)+average(163,67,43,19)

    Recall that
    average(163,67,43,19) mod (1+2+3+7+11) = 73 mod 24
    is the lowest prime congruent to 1 mod 24.

    1333 = 1332+1 make chron nom IC mist eerie is no longer “what cure US$” buy bore US slink UN PR Joe Act:

    2432=4*(1333-4*163-73)=4*608=4*(1333-4*163-average(163,67,43,19))
    608=1333-4*163-73=1333-4*163-average(163,67,43,19)

    Remember WHO’s at UN, as the throat of freedom is gnashed buy big te[a]ch media’s “demO CR at IC” transform m[a]ze UN:
    average(836,496) = (1333-1)/2 = 6 lyons (“minimal faithful representation”)

    Homeless “home” land leave $0[]well the freeze $in luck D-own.
    By “just 1” constitution of 1806, US mull a tory corrects UN method.

    “In God We Trust”

  115. Paul Vaughan says:

    Obvious to alert readers from the last comment, but pointing it out explicitly:

    φ(1333) = 73

    Throwing this a111literation in here “just” for puzzler fun:
    1333=1111+φ(1111)+φ(1111)=1111+111+111=φ(3333)

  116. Paul Vaughan says:

    “Hey lit a111 sister what have EU D-UN? WHO’s the only 1…?” (B111y11do1)

    Forest-level stuff OBviUS 2 alert readers “pickin’ up whatever’$Mayan” (Tom Petty) :
    Lucas no. 76 = 19 mod 24 + 43 mod 24 + 67 mod 24 + 163 mod 24 = 19+19+19+19 = 152/2 = 608/8 = 2432/32

  117. Paul Vaughan says:

    CRACK UN

    “Hit crews control…” — Tom Petty

    n = 3: {3, 4, 5; 6}
    27 = 3^3
    64 = 4^3
    125 = 5^3
    216 = 27+64+125 = ET(average(836,496))
    216 = 6^3 = ET((1333-1)/2)
    108=216/2=CT(162)=CT(ET(163))
    54=108/2=ET(162)=ET(ET(163))
    27=54/2
    “work kin on a mystery …that neverwood comm. tommy” — To:MP(ET)(TY)
    271=163+108=298-27

    1 = 1^2
    4 = 2^2
    9 = 3^2
    16 = 4^2
    25 = 5^2
    55 = 1+4+9+16+25

    784 = 28^2
    729 = 27^2
    55=784-729

    TP “go win wherever IT leads:”
    271=216+55

    Every Christmas the family “put together a puzzle”.

    Remember the George Polya method:
    Solve a simpler “problem”.

    s(4*163) = 70^2 / Σ(Heegner numbers mod 24) + 149 + 298 = 496 = s(496) = 1 / 4 * 1984
    70^2 = 1^2 + 2^2 + 3^2 + … + 22^2 + 23^2 + 24^2
    100 = 1+2+3+7+11+19+19+19+19

    D-no. Hints $O “well” We Can’t a Ford Circles. IT’s “Not. the Perfect” Christmas gift for red nos. JSUN.
    70p is weird for all primes p ≥ 149

    Jan. YOU wary “sol ID D-air IT” $savagely threatens DARK, COLD WINTER nightmares FAR, FAR BEYOND THE WORST of George Orwell’s WORST luck D-own fears. I don’t think people even realize what Can. happen if IT (inverted totalitarianism) keeps pushing $0 savagely, but I do hope they volunteer to take a very, very long walk alone and solve the traditional family Christmas puzzles online for one “sporadic and exceptional” Christmas, so we don’t find “Joe’s Globe HELL KNEW YEAR Sin 1984“.

  118. Paul Vaughan says:

    In last comment this was opportunistic pivot (begging Alberta$0[]we’ll not give Joe ammunition) :
    “s(4*163) = 70^2 / Σ(Heegner numbers mod 24) + 149 + 298 = 496 = s(496) = 1 / 4 * 1984”

    Above that exactly matches earlier notes — generalized pythagorean means of equivalent expression, including “the zipper”. I explore wildly and sometimes later note sharp refinements. Junction 162 got precisely refined after a wild introduction. Probably few noticed after dismissing first impressions that floored expectations.

  119. Paul Vaughan says:

    HarmonIC mean lie U+N $ 0 [ ] well:
    111=Φ(323)-(47+59+71)=Φ(196883-196560)-(47+59+71)

  120. Paul Vaughan says:

    PERFECT Jupiter-Saturn Framing — Part I

    First, we frame Jupiter-Saturn for exploration using the generalized Bollinger (1952) method. This part’s review.

    Beat Period

    19.8650360864628 = (29.4474984673838)*(11.8626151546089) / (29.4474984673838 – 11.8626151546089)
    9.93251804323141 = 19.8650360864628 / 2
    4.9662590216157 = 19.8650360864628 / 4

    Axial Period

    8.4561457463176 = (29.4474984673838)*(11.8626151546089) / (29.4474984673838 + 11.8626151546089)
    4.2280728731588 = 8.4561457463176 / 2
    2.1140364365794 = 8.4561457463176 / 4

    Harmonic Mean

    16.9122914926352 = (29.4474984673838)*(11.8626151546089)/((29.4474984673838+11.8626151546089)/2)

    Orientation

    61.0464822565173 = slip(29.4474984673838,11.8626151546089)
    835.546575435631 = slip(61.0464822565173,19.8650360864628)
    Note: 836 is the smallest untouchable weird number.

    Split Harmonic Mean Orientation

    right side:
    17.2616851219298 = (835.546575435631)*(16.9122914926352) / (835.546575435631 – 16.9122914926352)
    8.63084256096492 = 17.2616851219298 / 2
    4.31542128048246 = 17.2616851219298 / 4

    left side:
    16.5767613988929 = (835.546575435631)*(16.9122914926352) / (835.546575435631 + 16.9122914926352)
    8.28838069944647 = 16.5767613988929 / 2
    4.14419034972324 = 16.5767613988929 / 4

  121. Paul Vaughan says:

    PERFECT Jupiter-Saturn Framing — Part II

    Next, we derive base-level slip cycles:

    right:
    131.716392653884 = slip(19.8650360864628,17.2616851219298)
    65.8581963269421 = slip(19.8650360864628,8.63084256096492)
    50.0715412599931 = slip(19.8650360864628,4.31542128048246)

    left:
    100.143082519986 = slip(19.8650360864628,16.5767613988929)
    50.0715412599931 = slip(19.8650360864628,8.28838069944647)
    96.1829470900285 = slip(19.8650360864628,4.14419034972324)

    From the base we review familiar combinations:

    104.443321929454 = slip(50.0715412599931,19.8650360864628)
    178.266850068779 = slip(65.8581963269421,9.93251804323141)
    208.886643858908 = slip(65.8581963269421,19.8650360864628)
    356.533700137559 = slip(131.716392653884,19.8650360864628)

    504.413226524327 = slip(131.716392653884,9.93251804323141)
    504.413226524329 = slip(131.716392653884,104.443321929454)
    504.413226524326 = slip(178.266850068779,131.716392653884)
    504.413226524325 = slip(208.886643858908,131.716392653884)

  122. Paul Vaughan says:

    PERFECT Jupiter-Saturn Framing — Part III

    Finally, we focus on the PERFECT connection to Heegner numbers:

    304.031799474187 = slip(96.1829470900285,9.93251804323141)

    608.063598948353 = slip(50.0715412599931,4.9662590216157)
    608.063598948353 = slip(100.143082519986,4.9662590216157)
    608.063598948375 = slip(96.1829470900285,19.8650360864628)
    608.06359894839 = slip(178.266850068779,65.8581963269421)
    608.063598948387 = slip(178.266850068779,104.443321929454)

    1216.12719789671 = slip(50.0715412599931,9.93251804323141)
    1216.12719789671 = slip(100.143082519986,9.93251804323141)
    1216.12719789674 = slip(96.1829470900285,50.0715412599931)
    1216.12719789676 = slip(104.443321929454,50.0715412599931)
    1216.12719789675 = slip(104.443321929454,96.1829470900285)
    1216.12719789679 = slip(178.266850068779,96.1829470900285)
    1216.1271978968 = slip(208.886643858908,50.0715412599931)
    1216.12719789679 = slip(208.886643858908,65.8581963269421)
    1216.12719789678 = slip(208.886643858908,96.1829470900285)
    1216.12719789678 = slip(208.886643858908,178.266850068779)
    1216.12719789679 = slip(356.533700137559,96.1829470900285)
    1216.12719789678 = slip(356.533700137559,131.716392653884)
    1216.12719789678 = slip(356.533700137559,208.886643858908)
    1216.12719789679 = slip(504.413226524325,208.886643858908)
    1216.12719789679 = slip(504.413226524325,356.533700137559)

    2432.25439579341 = slip(100.143082519986,19.8650360864628)
    2432.25439579349 = slip(100.143082519986,96.1829470900285)
    2432.25439579353 = slip(104.443321929454,100.143082519986)
    2432.25439579362 = slip(208.886643858908,100.143082519986)

    The aliquot sequence beginning at 608 ends at perfect number 496.
    s(608) = 652
    s(652) = 496
    s(496) = 496 = s(652) = s(s(608))

    s(608)=652=32+16+8+4+2+1+304+152+76+38+19
    19 is the only odd prime divisor of 608.
    19 = x mod 24 for x = 19, 43, 67, 163 ——- the top 4 Heegner numbers

    s(652)=496=4+2+1+326+163
    The only odd prime divisor of 652 is 163 —— the TOP Heegner number

  123. Paul Vaughan says:

    F2 or B : 178=2+3+5+7+11+13+17+19+23+31+47

    Comm. PR UN D-Comm. red?

  124. Paul Vaughan says:

    No. Comm. Meant 163178

    496=s(496)
    496=248+124+62+31+(16+8+4+2+1)
    496=(8+4+2+1+(1))*31 —— Remember this number: 31.

    Sum of prime divisors:
    652: 165 (22 Taxis Bunanajam)
    608: 21=42/2=84/4
    496: 33=66/2=132/4

    496=(163+2)*3+1
    652=(19+2)*31+1
    496=(31+2)*15+1

  125. Paul Vaughan says:

    PERFECT Jupiter-Saturn Framing — Encore

    323=196883-196560

    292=323-31=19+43+67+163
    354=323+31=2*(47+59+71)

    323 = 19+43+67+163 + 31
    323 = 2 * (47+59+71) – 31

    42 = 73 – 31
    104 = 73 + 31
    Sporadic and Exceptional

    s(496) = 496 = 16+8+4+2+1 + 248+124+62+31
    31 is the only odd prime divisor of perfect number 496.
    average(19,43,67,163) mod (1+2+3+7+11) = 73 mod 24 = 1

  126. Paul Vaughan says:

    194

    Ly : 163 = 2+3+5+7+11+31+37+67
    F1 or M : 378 = 2+3+5+7+11+13+17+19+23+29 +31 +41+47+59+71
    F2 or B : 178 = 2+3+5+7+11+13+17+19+23+31+47 —- recall 4270=s(4370)

    Only 1 of the odd supersingular primes (and more generally the odd sporadic simple group primes) is a divisor of perfect no. 496: 31.

    316 = 2+3+5+7+11+13+17+19+23+29 -31 +41+47+59+71
    Compare with M sum above to spot the bold ( + vs. – ) difference.

    Compare with Heegner sum:
    316 = 1+2+3+7+11+19+43+67+163

    Note well the lowest prime congruent to 1 mod 24:
    average(19,43,67,163) mod (1+2+3+7+11) = 73 mod 24 = 1

    Sporadic and Exceptional” … and perfectly D-visible:

    194 = 163 + 31 ———————————- superstring answers to McKey miss s(496)tory
    132 = 163 – 31 = Corbyn lookback period = 2 * 66

    100 = sum of Heegner numbers mod 24 = (2+3+7+19+19)+(1+11+19+19) = 50+50

    104 = 73 + 31 = 4 * 26
    42 = 73 – 31

    Scale to squarely devise some supersupersingular monster group differing perfectly from Mayan level 5 calendar:

    26 = divisor^2 sum for 5
    130 = 378 – 248 = divisor^2 sum for 10
    260 = 2*378 – 496 = divisor^2 sum for 15

    Perfect number 28 = s(28) frames 56 faces, 84 edges, and 24 vertices.
    harmonic mean (28,84) = 42 = 66 – 24 = beat period of (28,84)
    axial period of (56,42) = 24 = 66 – 42 — Table 2

  127. Paul Vaughan says:

    100

    1 = 1 mod 24
    2 = 2 mod 24
    3 = 3 mod 24
    7 = 7 mod 24
    11 = 11 mod 24
    19 = 19 mod 24
    19 = 43 mod 24
    19 = 67 mod 24
    19 = 163 mod 24

    100 = (1+2+3+7+11)+(19+19+19+19) = average(96,104)
    96 = 163-67 = 104+59-67 = 37+59 = s(323)+59 = s(196883-196560)+59

    52 = 76-24 = (19+19+19+19)-(1+2+3+7+11) = 104 / 2
    100-52 = 2*(1+2+3+7+11) = 48 = 2 * 24 = 96 / 2

    100 = 4370-4270
    271=196883-196560-52=323-52=496-194-31

    1=ROUND(19/24,0)
    2=ROUND(43/24,0)
    3=ROUND(67/24,0)
    7=ROUND(163/24,0)
    42=ROUND(19/24,0)*ROUND(43/24,0)*ROUND(67/24,0)*ROUND(163/24,0)=1806/43
    =489426/271/43

    11 may look left out …but IT (as in inverted totalitarianism) has a place in this:
    10=11-1
    9=11-2
    8=11-3
    4=11-7
    $0[]well D-UN McKay!
    They sum to the only odd prime divisor of s(496)=496 : average(19,43) = 496/16 = 31

    Noel that 11 makes a difference notable in Heegner numbers mod 24.
    4*(19-11)=4*8=32

    992=31*32=2*496 ——————————————– superstring 61 tip $5: “call ’em 11!”

    31=(11-MOD(1,24))+(11-MOD(2,24))+(11-MOD(3,24))+(11-MOD(7,24))
    32=(MOD(19,24)-11)+(MOD(43,24)-11)+(MOD(67,24)-11)+(MOD(163,24)-11)

    31=44-MOD(1,24)-MOD(2,24)-MOD(3,24)-MOD(7,24)
    32=MOD(19,24)+MOD(43,24)+MOD(67,24)+MOD(163,24)-44

    156=992-836=652-496=132+24=132+(31-7)=132+(489-465) (see below)
    =100+2*28=(589-489)+(63-7) ——- obvious with large & small divisor partitions below memorized

    s(496)
    =496=465+31
    =248+124+62+31
    +1+2+4+8+16

    s(608)
    =652=589+63
    =304+152+76+38+19
    +1+2+4+8+16+32

    s(652)
    =496=489+7
    =326+163
    +1+2+4

    Lots of things simply fit a perfect frame.

    Review notes on generalized pythagorean theorem above:
    216 = 24*(FLOOR(19/24,1)+FLOOR(43/24,1)+FLOOR(67/24,1)+FLOOR(163/24,1))
    We’ll come back to this after reviewing almost-integer generation (the method I formulated to mimic Ramanujan’s “tricks”).

  128. Paul Vaughan says:

    Heegner McKey’s Free Will

    God knows we can learn perfect fits weather left or right:

    496=(196883-196560)+2*(47+59+71)-(19+43+67)-104/2
    323=(196883-196560)
    37=s(196883-196560)

    323 = d(2,1/3,59) = R(2,1/3,59) – R(1,1/3,59)
    104 = d(2,1/2,58) = R(2,1/2,58) – R(1,1/2,58)
    104 = d(2,1/2,37) = R(1,1/2,37) – R(2,1/2,37)

    47 = d(5,1/5,54) = R(1,1/5,54) – R(5,1/5,54)
    71 = d(3,1/5,54) = R(1,1/5,54) – R(3,1/5,54) = d(2,1/2,9) = R(1,1/2,9) – R(2,1/2,9)
    225 = d(4,1/5,54) = R(4,1/5,54) – R(1,1/5,54) = d(3,1/2,9) = R(1,1/2,9) – R(3,1/2,9)

    54+108=φ(162)+Φ(162)=Φ(163)=162

    225=2*(47+59+71)-(19+43+67)=Φ(Φ(323))+(19+43+67)=194+31=15^2=496-271

    59=average(47,71)

    Split the 9 Heegner numbers into 2 groups and simply OBserve.

    73=average(19,43,67,163) = divisor^3 sum for 4 = 4^3+2^3+1^3 = 64+8+1
    24=1+2+3+7+11
    73 mod 24 = 1 ——- 73 is the lowest prime congruent to 1 mod 24
    1 = average(163,67,43,19) mod (11+7+3+2+1)

    Remember the paper “Sporadic and Exceptional” was written by McKay & He.

    152=225-73
    152 = d(2,1/2,25) = R(2,1/2,25) – R(1,1/2,25)

    194=152+42
    298=225+73=104+152+42=104+194
    323=298+25=1984-1728+42

    111=Φ(323)-(47+59+71)

    1333 = 32*( x mod (11+7+3+2+1) ) + 4*163 + average(163,67,43,19) where x = 163,67,43,19
    1333=AVERAGE(19,43)*AVERAGE(19,67)
    1332=divisor^3 sum for 11=496+836=12*111

    31=AVERAGE(19,43)=2^5-1
    992 = σ(25)*σ(σ(25)) = 31*32 = 2*496 = 2*s(496) —————– see $5 tip in last comment

  129. Paul Vaughan says:

    216 & 55

    271=163+108=163+CT(162)=163+CT(ET(163))=489426/43/7/3/2=489426/1806

    21 = sum of prime divisors of 608 = 19+2
    292 = 271+21
    292 = 163 + 67 + 43 + 19
    76 =163 mod (11+7+3+2+1) + 67 mod (11+7+3+2+1) + 43 mod (11+7+3+2+1) + 19 mod (11+7+3+2+1)
    76 = 55+21

    55=378-323 = sum of supersingular primes minus s(196883-196560)
    55=1^2+2^2+3^2+4^2+5^2
    55=28^2-27^2
    55=AVERAGE(43,67)

    323=196883-196560=378-55

    216=292-76
    216=271-55

    216=3^3+4^3+5^3
    216=6^3
    216=378-162=378-ET(163)
    ET(163)=162=378-216=4*129-2*177=2*(2*(19+43+67)-(47+59+71))
    216 = 316 – 100 = sum of heegner minus sum of each heegner number mod 24
    216 = 24*(FLOOR(19/24,1)+FLOOR(43/24,1)+FLOOR(67/24,1)+FLOOR(163/24,1))

    378=316+31+31=248+σ_2(10)=248+5*σ_2(5)=(496+σ_2(15))/2=(496+10*σ_2(5))/2
    —————————————————————————————————- Mayan calender

    144=163-19
    144=12^2
    144=59^2-196883/59
    0=59^2-196883/59-163+19
    177=59^2-196883/59+163-19-111
    111=59^2-196883/59+163-19-(47+59+71)
    111=59^2-196883/59+(163-19)-(47+59+71)
    111=59^2-196883/59+(163-163mod24)-(47+59+71)

    67=(47+59+71)-(19+43+67)+19
    59=(47+59+71)-(19+43+67)+11

    67=(47+59+71)-(19+43+67)+67mod24
    59=(47+59+71)-(19+43+67)+59mod24

    260 = σ_2(15) = 15^2 +5^2+ 3^2 +1^2 = 225 +25+ 9 +1 = 225+35
    130 = σ_2(10) = 10^2 +5^2+ 2^2 +1^2 = 100 +25+ 4 +1 = 129+1
    = 5^3 + 5 = 125 + 5 (225-100) + (9-4)

    Something I suspect few would realize:

    The perpendicular sum 125 turns up more times than any other (1728/2) while scrambling the 3 components of each of 24, 47, 59, 71, & 177 in 5*3 blocks of the 15 supersingular primes.

    I suggest Mayan sages knew what Norton calls “the voice of God” and the connection to Heegner numbers. They wouldn’t need computers. As you can see the math is simple. There’s just a lot of it.

    390=260+130
    390-6^2=354=2*177
    225=354-129
    177=47+59+71
    129=19+43+67

  130. oldmanK says:

    Re video of N Scafetta posted on other thread (link to video https://www.youtube.com/watch?v=bW5-h9wn3OQ )

    Interesting: at 16:23, mention of the 60 yr and 20 yr cycles. Kepler trigon?; the latter possibly moon cycle.
    But more interesting at 44:49 see displayed on board the words ‘983 yr cycle’. The appellation of millennial cycle is misleading; it is specifically the Eddy cycle. Over a period of 8000 yrs that cycle correlates nicely (more like ‘in an ugly way’) with great historical changes.

  131. Paul Vaughan says:

    He Ignore Numbers Bro

    The solar cycle length thread (a worthwhile diversion but with significant cost) stole time available for clarifying Heegner connections.

    DC-ID-ed. to post CRude math by today with out organizing to ease digestion.

    We’re left rightly imagine in clear simple connect shh! UN between Heegner numbers and simple sporadic groups as a secret of what curry US sly miss story US trades.

    Better west turn math education up exponentia11y without delay to match the comprehensively superior Chinese math education system. D-C’s UNs ABout west turn luck D-own PUBLIC math education were left up to those hood (WHO’s at UN?) $0[]well sabotage.

    In private we know “the cause” of west stern backlash ON populism is deeply rooted in poor math education that D-evil ops eXpeRt deepen D-ants-C.

  132. Paul Vaughan says:

    The Owl on the Signpost

    oldmanK wrote Dec. 21, 2020 “mention of the 60 yr and 20 yr cycles. Kepler trigon?”

    My yen call UN door well! tee hee hee he ignore bro

    Some say important communication has to be repeated:

    Only 1 of the odd supersingular primes (and more generally the odd sporadic simple group primes) is a divisor of perfect no. 496: 31.

    table 2 — p.9

    “spin-offs” of 20 & 60

    Seasons Best

  133. Paul Vaughan says:

    “McKey’s Perfect” Sol’s Test

    small list odd prime divsor well chain sov. s(496) = 496 = s(652) = s(s(608)) sci11UNs

    50=31+19=194-144
    144=163-19
    194=163+31

    recall D-for UN sov. squares
    59*(59-12)*(59+12)

    CR is mess $0 merry:
    Found door well!

  134. Paul Vaughan says:

    Perfect Mayan Supersingle Days =

    260 = sum of supersingular primes each mod 59
    = 248+12 = 496/2+(71-59) = (496+71-47)/2 = 2*(59+71) = 2*130

    This keeps getting easier every step of the way.
    The easily remembered symbols are a key — a key from which to build other things which are less easy to remember.
    The thought processes about how to organize for reliable memory were presumably different without computers (duh!)

  135. Paul Vaughan says:

    “Dough!” more my yen$SIM Sun

    super sing yule lure sum 378 plus or minus (59+average(47,71))
    sure lock average(260,496) (my UN,perfect) SIM MET try

  136. Paul Vaughan says:

    oldmanK wrote “see displayed on board the words ‘983 yr cycle’. The appellation of millennial cycle is misleading”

    980 & 1470 are a source of ill-founded mainstream hubris. Difficult communication is on the horizon. That is all I can say for now.

  137. oldmanK says:

    PV says “980 & 1470 are a source of ill-founded mainstream hubris.”

    Pls allow me to point to something ‘definite’. Pls look here: https://melitamegalithic.wordpress.com/2020/05/31/searching-evidence-keplers-trigons-and-events-in-the-holocene-2/
    ( or at that site, the posts: Searching Evidence -2; Searching Evidence -3; Searching Evidence – 4: Prehistoric Mass Burials. )

    Do note that the 980yr Eddy cycle, from 6150bce to ~1700ce appears to have produced some effect on the Earth, without missing a step. Specifically at the roots. The source of the Eddy curve was from https://judithcurry.com/2018/06/28/nature-unbound-ix-21st-century-climate-change/ fig 122 The dates were not fitted to the Eddy curve, they just fell in place; had already been established some years before.

    The many number sequences above are to me somewhat alien, though I think I perceive, perhaps mistakingly, resonance points. What is the 980yr period? The response date is never so accurate. The strain release or fracture point of a non homogeneous body like earth (or a metal alloy) is never precise. The failure point can be early or late from a driving force that also peaks cyclically. Or a changing driving frequency on an object with a more stable resonant response? Or ???? (multiple forcing frequencies of varying power, acting on a body with multiple resonant points ??).

    Whatever, the evidence for the response is there; clear; in several-to-many unrelated proxies. I do not think one should lose sight of that fact.

  138. Paul Vaughan says:

    oldmanK,

    I have isolated the unification bridge. Standby.

  139. oldmanK says:

    PV thank you. Take your time, and make the best of the coming days. Whatever, stay safe, and peace to you and all the rest here.

  140. Paul Vaughan says:

    Seasons Best Tabulation

    0	1	2	3	4	5	6
    11.8626151546089	11.8548387096774	11.8624919302776	11.8625505633652	11.8626091970325	11.8626108257537	11.862621031421
    29.4474984673838	29.4	29.4471153846154	29.4474766943004	29.4474766943004	29.4474820471031	29.4475131572382
    19.8650360864628	19.8648648648649	19.8648648648649	19.8648648648649	19.8650292882492	19.8650314196813	19.865045881586
    8.4561457463176	8.44827586206897	8.45605154164749	8.45611112940712	8.45614092360186	8.45614219261852	8.45614994390406
    16.9122914926352	16.8965517241379	16.912103083295	16.9122222588142	16.9122818472037	16.912284385237	16.9122998878081
    61.0464822565173	61.25	61.046511627907	61.0449588854935	61.046511627907	61.0465087523851	61.046511627907
    835.546575435631	734.999999999992	835.227272727272	836.100189828859	835.518042500587	835.523429035292	835.547397485995
    2432.25439579341	1469.99999999966	2450.00000000031	2400.9799918338	2433.85038856765	2433.5510445144	2432.23243052504
    1216.12719789676	735.000000000099	1224.99999999973	1200.48999591718	1216.92519428372	1216.77552225649	1216.11621526286
    1216.12719789678	1469.99999999953	1224.99999999967	1200.48999591714	1216.92519428368	1216.7755222565	1216.11621526286
    608.06359894839	734.999999999766	612.499999999834	600.244997958568	608.46259714184	608.387761128249	608.058107631428
    608.06359894838	367.50000000005	612.499999999865	600.244997958589	608.462597141861	608.387761128246	608.058107631431
    504.413226524325	489.999999999987	503.424657534283	506.185274140598	504.323847956661	504.340628057757	504.414770415359
    504.413226524327	294.000000000008	503.424657534281	506.1852741406	504.323847956663	504.340628057757	504.414770415356
    504.413226524326	1469.99999999953	503.424657534283	506.185274140597	504.323847956662	504.340628057757	504.414770415356
    356.533700137559	489.999999999987	356.796116504845	356.055055315709	356.557582078734	356.553118735246	356.533527513542
    208.886643858908	183.750000000002	208.806818181821	209.025047457214	208.879510625147	208.880857258827	208.886849371497
    178.266850068779	244.999999999993	178.398058252422	178.027527657855	178.278791039367	178.276559367623	178.266763756771
    131.716392653884	133.636363636364	131.720430107527	131.70597313467	131.716815566777	131.716741939305	131.71645080763
    104.443321929454	104.999999999999	104.40340909091	104.512523728607	104.439755312573	104.440428629413	104.443424685748
    100.143082519986	97.9999999999997	100.136239782016	100.152954682491	100.142507215838	100.142619417789	100.143139752742
    96.1829470900285	105.000000000002	96.2041884816757	96.142526390143	96.184910583203	96.1845465194682	96.1829655363887
    65.8581963269421	66.8181818181818	65.8602150537634	65.8529865673348	65.8584077833885	65.8583709696525	65.8582254038149
    50.0715412599931	49.0000000000001	50.0681198910083	50.0764773412455	50.0712536079191	50.0713097088945	50.0715698763709
    
  141. Paul Vaughan says:

    Seasons Best Merger: 1470 & 2400

    4900 = 70^2 = 1^2 + 2^2 + 3^2 + … + 22^2 + 23^2 + 24^2

    Recall:
    1. the 2400 thread.
    2. two 1470 threads.
    3. U’s at 1/2 while others whole (as in whole number) with Laskar’s harmonic mean tuning.

    2940 = (1470)*(980) / (1470 – 980)
    993.243243243243 = (73500)*(980) / (73500 – 980)
    1185.48387096774 = (1470)*(993.243243243243)/((1470+993.243243243243)/2)
    1986.48648648649 = (2940)*(1185.48387096774) / (2940 – 1185.48387096774)
    844.827586206896 = (2940)*(1185.48387096774) / (2940 + 1185.48387096774)
    1689.65517241379 = (2940)*(1185.48387096774)/((2940+1185.48387096774)/2)
    Divide those by 100 and study column 1 slip cycles comparatively.

    Rudolph: “That’s the crude intro …”

    1470 = (1470)*(735) / (1470 – 735)
    490 = (1470)*(735) / (1470 + 735)
    980 = (1470)*(735)/((1470+735)/2)
    980 = (73500)*(993.243243243243) / (73500 + 993.243243243243)

    “… to ‘Santa’s quick’ red nos. informal fog.”

  142. Paul Vaughan says:

    Seasons Best Controversy

    Have no. con tact.
    Give no. Am. UN IT shh!own.

    God nos. IT: on “day won We’11” sea dawns 19th “hindsight is 1984”.

    0: Original (Seidelmann 1992)
    2432.25439579341

    1: UNstable CRude Est Worldview (73500)
    1469.99999999966

    2: crude find tuning 73500/4
    2450.00000000031
    11.8624919302776 = (18375)*(11.8548387096774) / (18375 – 11.8548387096774)
    29.4471153846154 = (18375)*(29.4) / (18375 – 29.4)

    3: UNsharp source SOV. CONTROVERSY !
    2400.9799918338
    11.8625505633652 = (2400000)*(11.8624919302776) / (2400000 – 11.8624919302776)
    29.4474766943004 = (2400000)*(29.4471153846154) / (2400000 – 29.4471153846154)

    4: SIMPLY sharpUN no. controversy!!
    2433.85038856765
    11.8626091970325 = (1200000)*(11.8624919302776) / (1200000 – 11.8624919302776)
    29.4474766943004 = (2400000)*(29.4471153846154) / (2400000 – 29.4471153846154)

    5: La2011 Table 6 Reign D-air !!!!!
    2433.5510445144
    11.8626108257537 = (1183561.64383561)*(11.8624919302776) / (1183561.64383561 – 11.8624919302776)
    29.4474820471031 = (2364963.50364963)*(29.4471153846154) / (2364963.50364963 – 29.4471153846154)
    1/(s4-s3) =
    1183561.64383561 = (72993.5229512813)*(68753.3156498674) / (72993.5229512813 – 68753.3156498674)
    1/(g4-g3) =
    2364963.50364963 = (74619.9907876555)*(72337.575351641) / (74619.9907876555 – 72337.575351641)

    6: Exploring symmetrically balanced residuals:
    2432.23243052504
    11.862621031421 = (1090000)*(11.8624919302776) / (1090000 – 11.8624919302776)
    29.4475131572382 = (2180000)*(29.4471153846154) / (2180000 – 29.4471153846154)

    in 2020 We
    CO[V]well IT’s
    CO[2]IDoorwell
    WHO’s at UN.

    Bye don19 NY 8:4 warn in buy D-UN mess age:
    “Hear comms. year 19th nerve US brake D-own” — Roll UN Stones

    CRews[.4]miss story red some ware Norton D-scribe monster US moonsh!nos. “the voice of God”.

    IN GOD WE TRUST

  143. Paul Vaughan says:

    Seidelmann (1992) Table 5.8.1 matches sidereal periods listed here.
    ALERT: Table 15.6 says “sidereal” but is off from this while matching quantities listed as “tropical” on pages in this series. Table 15.6 has propagated into other books. The quantities for Earth are an immediate tell-tale sign.

  144. Paul Vaughan says:

    F(IT) Faith in-Eur. Favor IT Orrery

    Years ago while studying and teaching stats and living in university residence I had a memorable late-night laundry room conversation with a beautiful math grad student.

    She expressed respect for dealing with messy real-world data and underscored that she chose math because it’s clean, pure, and stable.

    3 or 4 years ago I began a boycott of solar and climate exploration (briefly interrupted recently), motivated by (a) incromprehensibly inappropriate and excessive harassment in solar and climate discourse (whether founded in ignorance or deception intolerably dark either way) and (b) intransigently pesky data manipulation.

    Readers will note that I switched to expressing cynicism while exploring stable math (where the goalposts aren’t being moved politically). I had come to appreciate the perspective of the bright young lady I met in the laundry room years earlier.

    Little did I realize the convergence of streams that would result. I suspect some readers will not fathom the questions raised by the “Seasons Best” trilogy.

  145. Paul Vaughan says:

    Hindsight 2020 in Light of Seidelmann Typo — Part I

    Me, V, E, Ma beats with Seidelmann (1992) tables 5.8.1 & 15.6:

    25811.3691836373 = (0.240846697327135)*(0.24084445) / (0.240846697327135 – 0.24084445)
    25757.05496809 = (0.615197263396975)*(0.61518257) / (0.615197263396975 – 0.61518257)
    25763.987503107 = (1.00001743371442)*(0.99997862) / (1.00001743371442 – 0.99997862)
    25902.4692610417 = (1.88084761346252)*(1.88071105) / (1.88084761346252 – 1.88071105)

    Compare:

    25808.590009763 = harmean(25811.3691836373, 25757.05496809, 25763.987503107, 25902.4692610417)
    25808.3716263561 — tight match with previously noted Earth-Moon slip cycle (look near end of linked comment)
    -0.000846165586 = % error

  146. Paul Vaughan says:

    Hindsight 2020 in Light of Seidelmann Typo — Part II

    J, S, U, & N:

    23094.6280196893 = (11.8626151546089)*(11.85652502) / (11.8626151546089 – 11.85652502)
    36133.4834429379 = (29.4474984673838)*(29.42351935) / (29.4474984673838 – 29.42351935)
    26114.2236547808 = (84.016845922161)*(83.74740682) / (84.016845922161 – 83.74740682)
    25259.6956047047 = (164.791315640078)*(163.7232045) / (164.791315640078 – 163.7232045)

    Note carefully Saturn’s special weigh with numbers.

    25759.505729618 = (1+MOD(29.42351935,1)+1+1) / (1/23094.6280196893 + MOD(29.42351935,1)/36133.4834429379 + 1/26114.2236547808 + 1/25259.6956047047)
    25763.987503107 — compare
    -0.017395496285 = % error

    25811.0487949114 = (1+MOD(29.4474984673838,1)+1+1) / (1/23094.6280196893 + MOD(29.4474984673838,1)/36133.4834429379 + 1/26114.2236547808 + 1/25259.6956047047)
    25808.590009763 — compare
    0.009527003015 = % error

  147. Paul Vaughan says:

    2020 Tropical Rabbit Sequence

    1.59868953279706 = (0.99997862)*(0.61518257) / (0.99997862 – 0.61518257)
    0.761743683794994 = (0.99997862)*(0.61518257)/((0.99997862+0.61518257)/2)
    0.814043357147551 = (11.85652502)*(0.761743683794994) / (11.85652502 – 0.761743683794994)
    22.1348191900864 = slip(1.59868953279706,0.407021678573776)
    22.1348630064402 = (5^(1/2)+1)*(7-1)/(1/((7^0)+1/((7^1)+1/((7^1)+1/((7^2)+1/((7^3)+1/((7^5)+1/(7^8))))))))
    -0.000197951773 = % error

  148. Paul Vaughan says:

    Links for review now that we see the rabbit sequence appearing most clearly in Venus-Earth tropical context: _1_ & _2_

    “Perhaps even more remarkably […] we can replace the base 2 by any real number bigger than 1 […]”

    “This is a book I go back to so very often […] chapter 3 on Integer Functions has a marvellous section on Floors and Ceilings and spectra.”

    Concrete Mathematics (2nd edition, 1994) by Graham, Knuth and Patashnik, Addison-Wesley.

    k.[0,m).Σ ⌊(nk+x)/m⌋ = k.[0,n).Σ⌊(mk+x)/n⌋, integers m, n greater than 0
    last equation p.94 (pdf p.107)

    review:
    let n = 146, m = 208, & x = 0
    15008 = ⌊ 5482096 / 365.256367733331 ⌋ ——- where 5482096 is a famous Mayan number

  149. Paul Vaughan says:

    JEV sidereal review to compare and contrast with tropical:

    1.59868955949705 = (1.00001743371442)*(0.615197263396975) / (1.00001743371442 – 0.615197263396975)
    0.761766209372164 = (1.00001743371442)*(0.615197263396975)/((1.00001743371442+0.615197263396975)/2)
    0.814040387734912 = (11.8626151546089)*(0.761766209372164) / (11.8626151546089 – 0.761766209372164)
    22.1392314983837 = slip(1.59868955949705,0.407020193867456)
    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    -0.000000038239 = % error

    lim s→∞
    (φ^(2s)+1/s)^(e/s+1/(2s))
    = φ(φφ)^e —- the link is to Schneider’s classic proof

    Recall: 2, 12, & 240 are both highly and sparsely totient.

    Just replace φ with 240 in the calculation to switch from e to 1/E where E is Earth’s sidereal orbital frequency. 240 doesn’t need as many terms to converge.

    Forthcoming: simple exploration of 240 in the context of perfect numbers, Heegner numbers, supersingular primes, and generalized Pythagorean theorem.

  150. Paul Vaughan says:

    Symbolic Western Outreach

    The “Seasons Best” trilogy above demonstrates how ever-so-gentle, systematic orrery parameter tweaks switch between the “Charvatova 2400” and “1470” worldviews from a “Siedelmann’s Perfect” (as in perfect numbers) reference frame.

    Remember that 19 = x mod 24 where x = 19, 43, 67, & 163.
    We’re not done with Heegner numbers and monstrous moonshine.

    I suspect some will be studying the structure of the equations used in Seidelmann & Laskar.

    In God (but maybe knot SIM MET trick polynomials) We Trust

    Season’s Best everyone. Let’s eliminate western financial terrorism on homeland citizens. The pandemic has left some more wealthy and free while crushing $0[]well others. IT isn’t right.

  151. Paul Vaughan says:

    Substituting JSUN harmonic mean for JS harmonic mean:

    30.4320075307947 = (835.546575435631)*(29.3625733662892) / (835.546575435631 – 29.3625733662892)
    7.60800188269868 = 30.4320075307947 / 4

    208.880684806261 = slip(65.0170708690834,8.4561457463176)
    51.0763050439315 = slip(19.8650360864628,7.60800188269868)

    982.019421406588 = slip(208.880684806261,65.0170708690834)
    2331.75304290408 = slip(208.880684806261,51.0763050439315)

  152. Paul Vaughan says:

    171.406220601552 = (164.791315640078)*(84.016845922161) / (164.791315640078 – 84.016845922161)

    with JSUN harmonic mean

    35.4322620251755 = (171.406220601552)*(29.3625733662892) / (171.406220601552 – 29.3625733662892)
    8.85806550629388 = 35.4322620251755 / 4

    25.0682769387836 = (171.406220601552)*(29.3625733662892) / (171.406220601552 + 29.3625733662892)

    163.772511943922 = slip(35.4322620251755,19.8650360864628)
    81.8862559719609 = slip(19.8650360864628,8.85806550629388)

    95.7061647054583 = slip(25.0682769387836,19.8650360864628)
    280.639256382227 = slip(95.7061647054583,16.9122914926352)

    656.951889222652 = slip(280.639256382227,81.8862559719609) ————————————-
    979.858094471807 = slip(280.639256382227,163.772511943922) —————————————–

    14.7232679147118 = (11.8626151546089)*(6.56951889222652) / (11.8626151546089 – 6.56951889222652)
    29.4465358294237 = 2*14.7232679147118
    25808.9894765272 = (1+MOD(29.4465358294237,1)+1+1) / (1/23094.6280196893 + MOD(29.4465358294237,1)/36133.4834429379 + 1/26114.2236547808 + 1/25259.6956047047)
    0.001547805456 = % error

    5255.61511378122 = 8*656.951889222652
    5256 = ⌊5255.61511378122⌉ = 7920 – 2400 – 240 – 24 ————————– 7920 = M11 order
    3.14501624941799 = 5255.61511378122 / 835.546575435631 / 2

    14.7237492336919 = 29.4474984673838 / 2
    We’ve given this side plenty of attention…
    61.0464822565173 = (14.7237492336919)*(11.8626151546089) / (14.7237492336919 – 11.8626151546089)
    …but not the other:
    6.56961471832961 = (14.7237492336919)*(11.8626151546089) / (14.7237492336919 + 11.8626151546089)
    0.001458647196 = % error

    656.961471832961
    5255.69177466369
    3.14506212410931

  153. Paul Vaughan says:

    4270.09258127429 = slip(164.791315640078,84.016845922161)
    48590.8284812209 = slip(4270.09258127429,171.406220601552)

    29.3803273755345 = (48590.8284812209)*(29.3625733662892) / (48590.8284812209 – 29.3625733662892)
    7.34508184388362 = 29.3803273755345 / 4

    61.3371935567961 = slip(29.3803273755345,19.8650360864628)
    67.2332876489389 = slip(19.8650360864628,7.34508184388362)

    2734.13856809656 = slip(67.2332876489389,16.9122914926352)
    6441.4623444278 = slip(2734.13856809656,61.3371935567961)
    25765.8493777112 = 4*6441.4623444278

    5221.17476429629 = slip(2432.25439579341,131.716392653884)
    25765.8493777112 = 4 * 6441.4623444278
    20884.6990571851 = 4 * 5221.17476429629
    110242.867945166 = (25765.8493777112)*(20884.6990571851) / (25765.8493777112 – 20884.6990571851)

  154. Paul Vaughan says:

    Hindsight 2020 in Light of Seidelmann Typo — Part III

    Alternate weighting…
    8.45107360405992 = (29.42351935)*(11.85652502) / (29.42351935 + 11.85652502)
    0.418531083199004 = MOD(8.45107360405992*φ/4,1)
    …gives:
    25748.7187156079

    Alternate weighting…
    8.4561457463176 = (29.4474984673838)*(11.8626151546089) / (29.4474984673838 + 11.8626151546089)
    0.420582807841181 = MOD(8.4561457463176*φ/4,1)
    …gives:
    25753.1582384253

    25759.9581995811 = (1+5^2/59+1+1)/(1/23094.6280196893+5^2/59/36133.4834429379+1/26114.2236547808+1/25259.6956047047)

    25810.4394697625 = (1+1/√5+1+1) / (1/23094.6280196893 + 1/√5/36133.4834429379 + 1/26114.2236547808 + 1/25259.6956047047)

  155. Paul Vaughan says:

    Alternate Derivation of Sylvester’s Sequence

    Terse formulation + easier to grasp geometrically.

    harmonic means:
    UN:
    111.292543528394 = (164.791315640078)*(84.016845922161)/((164.791315640078+84.016845922161)/2)
    JS:
    16.9122914926352 = (29.4474984673838)*(11.8626151546089)/((29.4474984673838+11.8626151546089)/2)

    their beat:
    19.9428577113341 = (111.292543528394)*(16.9122914926352) / (111.292543528394 – 16.9122914926352)

    slip on JS frame:
    5090.68769455895 = slip(19.9428577113341,19.8650360864628)
    489425.980666482 = slip(5090.68769455895,8.4561457463176)

    489426 = 2*3*7*43*271
    489426 = 1806*271

    Absolutely fantastic and simple.

  156. Paul Vaughan says:

    Classic Review

    18.7636626447678 = (171.406220601552)*(16.9122914926352) / (171.406220601552 – 16.9122914926352)
    4.69091566119194 = 18.7636626447678 / 4

    338.432743555958 = slip(19.8650360864628,18.7636626447678)
    2311.47726940065 = slip(338.432743555958,4.9662590216157)

  157. Paul Vaughan says:

    I need to switch focus now. Someone else can be doing this type of work. It isn’t hard; there are just a lot of combinations to explore. Teamwork: It concerns me that others aren’t focused on monstrous moonshine and Heegner numbers. It makes me suspicious.

  158. oldmanK says:

    Something new (the idea resulting from the JS conjunction and the Croatia earthquake at the full moon).
    Major earthquakes Croatia :
    1667 Dubrovnik earthquake
    1880 Zagreb earthquake
    2020 Petrinja earthquake
    the gaps are multiple of 70. coincidence?????

  159. oldbrew says:

    oldmanK says: Coincidence?
    December 30, 2020 at 10:02 am
    – – –
    Perhaps, but 1880 and 2020 both in first year or so of low solar cycles (expected for current SC), and 1668 in the Maunder Minimum.

    https://en.wikipedia.org/wiki/List_of_solar_cycles

  160. Paul Vaughan says:

    alright 1 encore note gentlemen before resuming more fundamental attention:
    113.780000311666 = (19.8650360864628)*(16.9122914926352) / (19.8650360864628 – 16.9122914926352)
    208.09114070835 = (1216.12719789676)*(113.780000311666)/((1216.12719789676+113.780000311666)/2)
    same calculation with the “1470 worldview” gives 196 & 210 (for 735 & 1470 respectively)

    ———————————

    Strategic Parity

    104 = d(4,1/2,58) = R(4,1/2,58) – R(1,1/2,58)
    104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π
    a = 0.000034332275390625 = ⌊(e^√58π)⌉ – e^√58π
    395.999999581314 = (e^√58π)^(1/4)
    396 = ⌊(e^√58π)^(1/4)⌉
    b = 396 – 395.999999581314 = ⌊(e^√58π)^(1/4)⌉ – (e^√58π)^(1/4)
    82 = a / b
    186 = 104+82
    744 = 4*186
    496 = 2/3*744
    1984 = 4*496

    73=AVERAGE(42,104) ————- “Sporadic & Exceptional” indeed
    73=AVERAGE(19,43,67,163)
    104=73+31
    42=73-31
    194=163+31
    132=163-31

  161. Paul Vaughan says:

    The Mayan sages simply keyed the following with pythagorean theorem:

    50=31+19=194-144
    144=163-19
    194=163+31

    59*(59-12)*(59+12) = 196883
    12^2 = 144 = 59^2-196883/59

    323=196883-196650
    323-31=292=19+43+67+163
    354+31=354=2*(47+59+71)

    378=sum of supersingular
    316=sum of supersingular with sign of +31 flipped to -31
    316=sum of heegner

    271=292-31 —————————————————————— Sylvester Sequence

  162. Paul Vaughan says:

    “ally” UN count “Or” flaw lane

    162=ET(163)=66+96

    s(496-108)=s(496-CT(163))=298

    298=152+104+42
    s(298) = 152
    s(s(152)) = 118 = 2*59 = 47+71
    s(118) = 62 = s(2*59) = 2*31

    248 = 104 + 144 = 496/2
    186 = 144 + 42 = 744/4

    496 = s(s(298)) + sum of supersingular primes = 8 * s(s(s(298)))

    31=AVERAGE(19,43)

    589=1333-744
    589=608-19
    589=31*19
    589=19*496/16

    744=652+73+19=4*163+73+19=25*29+19

    There’s a reason no one noticed: it isn’t obvious (duh!)

    “But this has nothing to do with the sun…” — conservatives voices lost in 1984…

  163. Paul Vaughan says:

    While retaining hope that the number theory community will quickly and formally clarify for the world the link between Heegner numbers and monstrous moonshine, I do realize some observers here are looking for something else.

    Very well. This is crude but those in the know will easily recognize it nonetheless.

    JEV tropical and sidereal:

    111063.836252585 = (22.1392314983837)*(22.1348191900864) / (22.1392314983837 – 22.1348191900864)
    22.1370251243719 = (22.1392314983837)*(22.1348191900864)/((22.1392314983837+22.1348191900864)/2)
    490 = 22.1348191900864 * 22.1370251243719
    …980, 1470, 36750, 73500

    1.00002643794033 = (111063.836252585)*(1.00001743371442) / (111063.836252585 – 1.00001743371442)
    1.00002638193018
    0.000005600867 = % error

    There’s a lot more that can be said around this. Clean, simple stuff — surely widely known already, yet we haven’t seen it addressed (curiously).

  164. Paul Vaughan says:

    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    14.7237492336919 = 29.4474984673838 / 2
    13.1392294366592 = (14.7237492336919)*(11.8626151546089)/((14.7237492336919+11.8626151546089)/2)
    plus 9 equals:
    22.1392294366592
    -0.000009350777 = % error

    22.1392285019811 = 2 * 5090 / 744 * φ
    -0.000013572596 = % error

    Just putting in new combinations familiar numbers.

  165. Paul Vaughan says:

    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    22.1392314983837 = 1/(3/V-5/E+2/J)
    0.000000038239 = % error

    Replace φ with 240 in the expression for e here.

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)
    1.00001743371442 = 1/E
    0.000000018929 = % error

    0.61519726351855 = 3/(1/22.1392315068494+5/1.00001743390371-2/11.8626151546089)
    0.615197263396975 = 1/V
    0.000000030106 = % error

  166. Paul Vaughan says:

    So what’s left tune notice?
    992 = 1*2*16*31

    496 = 16*31

    Perfect sci11UNs comm. UN IC k? s(496)hh!…..

  167. Paul Vaughan says:

    Tropical UN slip-cycles match with sidereal JS slip-cycles:

    2432 &
    20884 = 4*5221

    171.444289533663 = (163.7232045)*(83.74740682) / (163.7232045 – 83.74740682)
    110.812300014126 = (163.7232045)*(83.74740682)/((163.7232045+83.74740682)/2)
    55.4061500070632 = (163.7232045)*(83.74740682) / (163.7232045 + 83.74740682)
    27.7030750035316 = 55.4061500070632 / 2

    3635.42278750964 = slip(163.7232045,83.74740682)

    29.3305236493692 = harmean(JSUN tropical periods)
    29.5690864909714 = (3635.42278750964)*(29.3305236493692) / (3635.42278750964 – 29.3305236493692)
    7.39227162274285 = 29.5690864909714 / 4

    891.227241339459 = slip(171.444289533663,7.39227162274285)
    20884.3849304411 = slip(891.227241339459,110.812300014126)

    849.121646024845 = slip(171.444289533663,29.5690864909714)
    2431.67883089196 = slip(849.121646024845,27.7030750035316)

  168. Paul Vaughan says:

    That clarifies the distinction from Laskar’s p = 25685 (follow the hyperlink in the last comment) :

    0.999978499031851 = (20884.3849304411)*(1.00002638193018) / (20884.3849304411 + 1.00002638193018)

    25683.9369504749 = (1.00001743371442)*(0.999978499031851) / (1.00001743371442 – 0.999978499031851)

    Compare with:
    25763.987503107 = (1.00001743371442)*(0.99997862) / (1.00001743371442 – 0.99997862)

  169. Paul Vaughan says:

    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    22.1348478401313 = (245/(φ^22+1/11)^(e/11+1/22)+√((245/(φ^22+1/11)^(e/11+1/22))^2+980))/2

    111789.639177999 = (22.1392315068494)*(22.1348478401313) / (22.1392315068494 – 22.1348478401313)
    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)

    1.00002637966847 = (111789.639177999)*(1.00001743390371) / (111789.639177999 – 1.00001743390371)
    1.00002638193018 = terrestrial anomalistic year period
    -0.000000245095 = % error ——————— 7 seconds per century

    111759.01908408 = (1.00002638193018)*(1.00001743371442) / (1.00002638193018 – 1.00001743371442)
    -0.027390815593 = % error

  170. Paul Vaughan says:

    The second line in the preceding comment is simply solving the quadratic equation that ties JEV tropical and sidereal to 4900 = 70^2 = 1^2 + 2^2 + 3^2 + … + 22^2 + 23^2 + 24^2 to check how tight the fit is (off by 7 seconds per century as noted).

  171. Paul Vaughan says:

    260 & 64000

    8.45305064436623 = (29.4474984673838)*(11.85652502) / (29.4474984673838 + 11.85652502)
    8.45416725812087 = (29.42351935)*(11.8626151546089) / (29.42351935 + 11.8626151546089)
    64000.2003304946 = (8.45416725812087)*(8.45305064436623) / (8.45416725812087 – 8.45305064436623)

    19.8619536084148 = (19.8759632567693)*(19.8479636956574)/((19.8759632567693+19.8479636956574)/2)

    11.8595693054519 = (11.8626151546089)*(11.85652502)/((11.8626151546089+11.85652502)/2)
    29.4355040251508 = (29.4474984673838)*(29.42351935)/((29.4474984673838+29.42351935)/2)
    19.8619536084148 = (29.4355040251508)*(11.8595693054519) / (29.4355040251508 – 11.8595693054519)

    19.8759632567693 = (29.42351935)*(11.8626151546089) / (29.42351935 – 11.8626151546089)
    19.8479636956574 = (29.4474984673838)*(11.85652502) / (29.4474984673838 – 11.85652502)
    19.8619536084148 = (19.8650360864628)*(19.8588720868409)/((19.8650360864628+19.8588720868409)/2)

    260 ~= 259.996348149018 = 5*φφ*19.8619536084148

  172. Paul Vaughan says:

    73=AVERAGE(19,43,67,163)
    73=365-19-43-67-163

    120=163-43
    240=163-19+163-67
    360=163-19+163-67+163-43

    129=19+43+67
    194=323-129
    194=163+31
    323=194+19+43+67
    163=194-31

    271=489426/43/7/3/2
    271=240+31
    209=240-31
    836=4*209

  173. Paul Vaughan says:

    Perfect Mayan Symmetry

    260=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71-(47+71)
    496=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71+(47+71)

    260=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71-(2*59)
    496=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71+(2*59)

    260=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71-(59+AVERAGE(47,71))
    496=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71+(59+AVERAGE(47,71))

    260=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71-2*196883/71/47
    496=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71+2*196883/71/47

  174. Paul Vaughan says:

    Earthly 240

    378=2+3+5+7+11+13+17+19+23+29+31+41+47+59+71
    316=2+3+5+7+11+13+17+19+23+29-31+41+47+59+71
    316=1+2+3+7+11+19+43+67+163

    240=1+2+3+7+11+19+43+67+163-MOD(19,24)-MOD(43,24)-MOD(67,24)-MOD(163,24)
    240=1+2+3+7+11+19+43+67+163-19-19-19-19
    240=1+2+3+7+11+19+43+67+163-76

    240=2+3+5+7+11+13+17+19+23+29-31+41+47+59+71-MOD(19,24)-MOD(43,24)-MOD(67,24)-MOD(163,24)
    240=2+3+5+7+11+13+17+19+23+29-31+41+47+59+71-19-19-19-19
    240=2+3+5+7+11+13+17+19+23+29-31+41+47+59+71-76

    76=MOD(19,24)+MOD(43,24)+MOD(67,24)+MOD(163,24)
    76=19+19+19+19

  175. Paul Vaughan says:

    [H]edge [O]ption (…and this saint nicholas stir red yen!)

    360 = (196883-196560) + s(196883-196560)
    360 = (323) + s(323) = 323 + 37
    “Walk like an Egyptian” — The Bangels
    104 = d(2,1/2,37) ~= 2*φφ*19.8619536084148

    mic on a11 scene at or[well 1984]
    speech so “free eu” can’t eve un safe flee warn all lies push un too herd

    “a11 the COP$in the dough nut$shhop” — the ban gel$
    “WA11 cli. con e.g. op$shh!un” — The B-angels
    “wei hua wei hua wei hua wei hua: woke lie con edge ops yen”

  176. Paul Vaughan says:

    163(Can’t Talk Key PR air)980

    Roads

    “Troubles will come and they will pass
    and don’t forget son there is someone up above”

    Fair

    Be a simple kind dove man…love and understand

    178.266850068779 = beat(131.716392653884,504.413226524325)
    178.266850068779 = beat(96.1829470900285,208.886643858908)
    178.266850068779 = beat(65.8581963269421,104.443321929454)
    178.266850068779 = axial(208.886643858908,1216.12719789676)
    178.266850068779 = harmean(104.443321929454,608.06359894839)
    178.266850068779 = harmean(96.1829470900285,1216.12719789676)

    Bridge

    “and if you do this IT’11 help you some sunny day” — Lynyrd Skynyrd “Simple Man”

  177. Paul Vaughan says:

    5256 = 73*72 = φ(1333)*Φ(111) = 7920-2400-240-24
    s(608) = 652; s(652) = 496; s(496) = 496

    36750.3379015986 = 8/(g_6+s_6+s_3+5*g_4+s_4+5*g_3+g_2+s_2) —– Laskar
    323 = 196883196560
    37 = s(323)

    Φ(1333) = 1260 = 608 + 652 = 608 + s(608) = d(2,1/2,33) = R(1,1/2,33) – R(2,1/2,33)

    993.252375718881 = 36750.3379015986 / 37
    496.626187859441 = 993.252375718881 / 2
    497 = 993 – 496 = 993 – s(652) = 993 – s(s(608))
    1333 = 497 + 836 = Φ(1333) + φ(1333) = 1260 + 73 = 608 + 652 + average(19,43,67,163)
    1332 = 496 + 836 = 1260 + 72 = Φ(1333) + Φ(111)
    1984 = 496 + (836 + 652) = 496 + (1488) = 496 + (3*496) = 4*496
    1987 = 2*497 + 993
    1986.50475143776 = 993.252375718881 * 2

    check balance:
    112 = 73 + 39 = φ(1333) + φ(111) —————– _L_ ; _J4_
    1444 = 1332 + 112 = 1333 + 111

    216 = Φ(Φ(1444)) = 3^3 + 4^3 + 5^3 = 6^3
    = sum of Heegner numbers minus sum of (Heegner numbers mod(1+2+3+7+11))
    = 19+43+67+163-19-19-19-19
    = 4*Φ(Φ(163)) = 2*φ(Φ(163))

    216 = 2432/2 – Φ(1111) = 2*608 – Φ(1111) = 1216 – 1000
    1333 = 1111 + φ(1111) + φ(1111) = 1111 + 111 + 111 = φ(3333)

    recall:
    111 = Φ(196883-196560)-(47+59+71)
    111 = Φ(323)-(47+59+71)
    111 = Φ(1260)-(47+59+71) = Φ(608+652)-(47+59+71) = Φ(608+s(608))-(47+59+71)
    111 = Φ(608)-(47+59+71)
    111 = Φ(365)-(47+59+71)

    all Heegner numbers appear in lines 2 & 3:
    196883/59 = 47*71 = 59^2-Φ(196883-196560)/2
    = s(19+43+67)*s(s(Φ(163)))
    = 59^2-((1+2+3+7+11)/2)^2

    47 = 59-12 = s(19+43+67)
    71 = 59+12 = s(67+67+67) = s(s(Φ(163)))
    24 = 71 – 47 = s(s(Φ(163))) – s(19+43+67) = 1+2+3+7+11

    576 = (1+2+3+7+11)^2 = 24^2
    288 = Φ(323) = Φ(1260) = Φ(608) = Φ(365)
    144 = ((1+2+3+7+11)/2)^2 = 12^2

    L: 163 = 652 / 4 = s(608) / 4 = 2+3+5+7+11+31+37+67

    113.757966840464 = slip(24.067904774739,19.8650360864628)
    113.760416666667 = (163)*(67) / (163 – 67)
    113.780000311666 = (19.8650360864628)*(16.9122914926352) / (19.8650360864628 – 16.9122914926352)

    1333 ~= (323)*(260) / (323 – 260)
    323 ~= (1333)*(260) / (1333 – 260)
    144 = 12^2 ~= (1333)*(130) / (1333 – 130)
    260 ~= (323)*(144) / (323 – 144)
    194 ~= (260)*(111) / (260 – 111)
    111 ~= (260)*(194) / (260 + 194) ~= (43)*(31) / (43 – 31)

    3337 = 47*71 = 196883/59
    292 = 19+43+67+163
    320.0013136289 = (3337)*(292) / (3337 – 292)
    64000.26272578 = 200*320.0013136289
    64000.2003306467
    0.000097492091 = % error

    The small distinction between 64000.26 & 64000.2 has diagnostic utility.

  178. Paul Vaughan says:

    There will be a series of comments on sporadic simple groups, 70^2 = 1^2 + 2^2 + 3^2 + … + 22^2 + 23^2 + 24^2, 260, & 64000 making hairline distinctions.

    Few will have the patience and interest to stay the course (e.g. $speed-chess$$ addicts$slaving destructively under the trolllionairfirehosepress$$sure of menacing political timetables$$).

    14.7237492336919 = 29.4474984673838 / 2
    13.1392294366592 = harmean(14.7237492336919,11.8626151546089)
    22.1392294366592 = 9 + 13.1392294366592
    22.134848529967 = = (245/22.1392294366592+√((245/22.1392294366592)^2+980))/2
    22.1370387665684 = harmean(22.1392294366592,22.134848529967)
    22.1370383309738 = (7920/√(2*64000.2003306175)
    0.000001967719 = % error

    490 = 22.1370387665684 * 22.134848529967
    64000.1978119299 = (7920/22.1370387665684)^2/2
    64000.2003306175
    -0.000003935437 = % error

    This alerts us to simple J & S tropical structure relative to J & S sidereal.
    It also helps clarify the structure of what’s on vs. near (but systematically off) the 70^2 frame.

    Years ago I knew I should investigate that suspicious list of numbers I shelved from Seidelmann (1992). There’s always an endless list of other things to explore — enough to consume many lifetimes.

    A half-dozen times over the years I thought about the list but focused on competing things. Not sure why but last month I dusted off the old file — and verified the simple typo (one that propagated into other published books).

    Since then insights are pouring like a string of domino’s. A good side-effect of the long delay: the temporary obstruction diverted me to first quantitatively master Bollinger’s (1952) symbolic method (eased intuitive link to Mayan & Egyptian methods and simple sporadic groups).

  179. Paul Vaughan says:

    Why $0men?

    Careful readers should already see the links between 22 & 70^2.

    “some peep hill ‘calm me’ ‘the bad apple’ but I may be ‘the sweet test’ APoll on the tree” — David Wilcox

    Note that there are 2 groupings of “22.137”, just as there are 2 groupings of “64000.2”.
    We’re drawing a diagnostic hairline distinction between the 2 groups.

    292 = 5256/18 = average(19,43,67,163)
    3337 = 47*71 = 196883/59

    320.0013136289 = beat(3337,292)
    22.1370275400459 = 7920 / √128000.52545156
    22.1370251243719 = harmean(22.1392314983837,22.1348191900864)
    -0.000010912368 = % error

  180. Paul Vaughan says:

    typo:
    “292 = 5256/18 = average(19,43,67,163)”

    correction:
    292 = 19+43+67+163
    73 = average(19,43,67,163)

    supplementary:
    657 = 3^2 × 73, the largest known number not of the form a^2+s with s a semiprime

    Rusty NASA JPL press realease:
    =
    It’s very puzzling […] At first, I totally didn’t believe it. It shouldn’t exist based on the conditions present on the Moon […]
    […]
    One major clue was the rust was more concentrated on the side of the moon that faces Earth — suggesting it was somehow linked to our planet.
    […]
    During these six days, Earth’s magnetic tail covers the moon’s surface with electrons, and all sorts of strange things can happen.
    =

  181. Paul Vaughan says:

    This one’s important.
    Earlier I introduced something left subtly imbalanced. Here I righten the balance:

    208.886643858907 = 1/(-8J+20S)
    208.886658253856 = beat(49000,207.999955892562)
    0.000006891272 = % error

    y = R(p,1/2,7) = 24.067904774739 = ⌊(e^√7π)^(1/p)⌉^p – e^√7π for p=2,3,4,6,12
    x = R(p,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/p)⌉^p – e^√58π for p=2,4
    z = -R(2,1/2,37) = 103.999977946281 = e^√37π – ⌊(e^√37π)^(1/2)⌉^2
    2z = 207.999955892562 = 2 * 103.999977946281

    11.8626151191159 = 1/((6/y+(1/4)/x)/5+(1/x+1/298)*φ^2)
    11.8626151546089 = 1/J
    -0.000000299201 = % error

    29.4474983942016 = 20/(1/(2z)+8*((6/y+(1/4)/x)/5+(1/x+1/298)*φ^2)-1/49000)
    29.4474984673838 = 1/S
    -0.000000248518 = % error

    2432.25097753771
    2432.25439579341
    -0.000140538576 = % error

  182. Paul Vaughan says:

    Using the sharpened J & S estimates (see last comment) :

    19.8650360202345 = beat(29.4474983942016,11.8626151191159)
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    -0.000000333392 = % error

    8.45614572224745 = axial(29.4474983942016,11.8626151191159)
    8.4561457463176 = axial(29.4474984673838,11.8626151546089)
    -0.000000284647 = % error

    16.9122914444949 = harmean(29.4474983942016,11.8626151191159)
    16.9122914926352 = harmean(29.4474984673838,11.8626151546089)
    -0.000000284647 = % error

    Review

    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    22.1392314983836 = 1/(3V-5E+2J) = 1/H —- Jupiter-Earth-Venus SIDEREAL
    0.000000038239 = % error

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)
    1.00001743371442 = 1/E
    0.000000018929 = % error

    H = 3V-5E+2J (algebra in frequency = 1/period)
    V = (1/3)H+(5/3)E-(2/3)J
    0.615197263582188 = 1/V = 1/((1/3)/22.1392315068494+(5/3)/1.00001743390371-(2/3)/11.8626151191159)
    0.615197263396975 = 1/V
    0.000000030106 = % error

  183. Paul Vaughan says:

    64000 & 260 Review

    Mayan sages embedded these symbols symbolically (probably a mnemonic device — e.g. “mind palace” = method of loci).

    16.9083345162417 = harmean(29.42351935,11.8626151546089) — S trop with J side
    16.9061012887325 = harmean(29.4474984673838,11.85652502) — S side with J trop
    128000.400661193 = beat(16.9083345162417,16.9061012887325)

    8.45416725812087 = axial(29.42351935,11.8626151546089) — S trop with J side
    8.45305064436623 = axial(29.4474984673838,11.85652502) — S side with J trop
    64000.2003304946 = beat(8.45416725812087,8.45305064436623)

    19.8650360864628 = beat(29.4474984673838,11.8626151546089) — S side with J side
    19.8588720868409 = beat(29.42351935,11.85652502) — S trop with J trop
    64000.2003306175 = beat(19.8650360864628,19.8588720868409)

    19.8619536084148 = harmean(19.8650360864628,19.8588720868409)
    52 ~= 51.9992696298035 = 1 * φφ * 19.8619536084148
    104 ~= 103.998539259607 = 2 * φφ * 19.8619536084148
    208 ~= 207.997078519214 = 4 * φφ * 19.8619536084148
    260 ~= 259.996348149018 = 5 * φφ * 19.8619536084148

  184. Paul Vaughan says:

    Now, pay careful attention as the hairline difference between the two “22.137” clusters emerges from 70^2 and propagates to the 2 corresponding “64000.2” clusters.

    Note well the MUCH larger %errors in this calculation chain (compared with those in this comment). They signal a systematic divergence of J, S, & JEV sidereal and tropical structure which we’re diagnosing.

    22.1348478401313 = (245/(φ^22+1/11)^(e/11+1/22)+((245/(φ^22+1/11)^(e/11+1/22))^2+980)^(1/2))/2
    22.1348191900864 = = 1/(3V-5E+2J) —- Jupiter-Earth-Venus TROPICAL
    0.000129434284 = % error

    22.1370394564725 = 490/((245/(φ^22+1/11)^(e/11+1/22)+((245/(φ^22+1/11)^(e/11+1/22))^2+980)^(1/2))/2)
    22.1370251243719 = harmean(JEV sidereal & tropical periods)
    0.000064742667 = % error

    490 = 22.1348478401313 * 22.1370394564725
    489.999048534373 = JEV tropical period * harmean(JEV sidereal & tropical periods)
    0.000194177036 = % error

    64000.193822779 = ( 7920 / 22.1370394564725 )^2 / 2
    64000.2766936711 = ( 7920 / 22.1370251243719 )^2 / 2
    -0.000129485209 = % error

    Monstrous moonshine and Heegner numbers together appear to define the offset.

    64000.193822779 = (7920 / 22.1370394564725)^2 / 2
    64000.2003306175 = long JS sidereal & tropical cycle (embedded in Mayan symbolism)
    -0.000010168466 = % error

    22.1370394564725 = 7920 / √(2*64000.193822779)
    22.1370383309738 = 7920 / √(2*64000.2003306175) = 7920 / √(128000.400661193)
    0.000005084233 = % error

    3337 = 47*71 = 196883/59 —————- _M_
    292 = 19+43+67+163 = 5256/18 ——— Heegner
    320.0013136289 = beat(3337,292)
    64000.26272578 = 200 * 320.0013136289
    64000.2766936711
    0.000021824740 = % error

    22.1370251243719 = 7920 / √(2*64000.2766936711)
    22.1370275400459 = 7920 / √(2*64000.26272578)
    -0.000010912368 = % error

    Almost-integers appear to minimize rotation (of frames relative to one another) where exact integer equality isn’t physically possible (because of competing minimizations).

  185. Paul Vaughan says:

    Review

    22.1392315068494 = (1.61803398874989^MOD(47+71,24)+1/MOD(59,24))^(2.71828182845905/MOD(59,24)+1/MOD(47+71,24))

    phi as monstrous aliquot trig function:
    φ = 2*cos(s(71^2)/s(59^2)/s(47^2)*8*π) where s(n) = aliquot sum for n
    φ = 2*cos(s((59+12)^2)/s(59^2)/s((59-12)^2)*8*π)
    Seriously: How many people know this familiar φ identity in monstrous aliquot form?

    71^2 = 5041; s(5041) = 72
    59^2 = 3481; s(3481) = 60
    47^2 = 2209; s(2209) = 48
    72/60/48 = (6/5/4)/12 = 1/40 = 0.025
    0.2 = 1/5 = 6/5/4/3*2 = 8/40
    φ = 2*cos(6/5/4/3*2π) = 2*cos(π/5)

    Here’s another one to mull: 240
    240 is both sparsely and highly totient.

    Schneider’s classic — alternate link: golden mobius totient

    I love this stuff.
    It’s fantastic.

    I remember years ago a video by a researcher named Cuk about “Jovian evection resonance” and now with dominoing insights after suddenly hurdling past the Seidelmann (1992) typo delay it looks like there’s fun work available for academics studying our solar system as a local example of number theory. By God’s gracioius guidance some bright mathematical spark(s) will (a) formally clarify Heegner number links to monstrous moonshine and (b) help peers see the classification theorem in beautifully succinct form.

    “Mathematics is the queen of the sciences—and number theory is the queen of mathematics.” — Gauss

  186. Paul Vaughan says:

    =
    Conway said of the monster group: “There’s never been any kind of explanation of why it’s there, and it’s obviously not there just by coincidence. It’s got too many intriguing properties for it all to be just an accident.” Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, “I can explain what Monstrous Moonshine is in one sentence, it is the voice of God.”
    =

    Better West Turn Math Education

    By God’s gracious guidance some bright mathematical spark(s) will (a) formally clarify Heegner number links to monstrous moonshine and (b) help peers see the classification theorem in beautifully succinct form.

    “Mathematics is the queen of the sciences—and number theory is the queen of mathematics.” — Gauss

  187. Paul Vaughan says:

    Comparing JS sidereal with JEV estimate based on φ & e:

    22.1370387665684
    22.1370394564725
    -0.000003116515 = % error

    64000.1978119299 = (7920/22.1370387665684)^2/2
    64000.193822779 = (7920/22.1370394564725)^2/2
    0.000006233029 = % error

    Comparing JS sidereal with Seidelmann (1992) JEV:

    22.1370387665684
    22.1370394536512
    -0.000003103770 = % error

    64000.1978119299 = (7920/22.1370387665684)^2/2
    64000.193839092 = (7920/22.1370394536512)^2/2
    0.000006207540 = % error

    Comparing JEV estimate based on φ & e with Seidelmann (1992) JEV:

    22.1392315068494
    22.1392314983837
    0.000000038239 = % error

    22.1348478401313 = (245/22.1392315068494+√((245/22.1392315068494)^2+980))/2
    22.1348478429522 = (245/22.1392314983837+√((245/22.1392314983837)^2+980))/2
    -0.000000012745 = % error

    22.1370394564725 = harmean(22.1392315068494,22.1348478401313)
    22.1370394536512 = harmean(22.1392314983837,22.1348478429522)
    0.000000012745 = % error

    490 = 22.1348478401313 * 22.1370394564725
    490 = 22.1348478429522 * 22.1370394536512
    0.000000000000 = % error

    64000.193822779 = (7920/22.1370394564725)^2/2
    64000.193839092 = (7920/22.1370394536512)^2/2
    -0.000000025489 = % error

  188. Paul Vaughan says:

    Demonstrating how 27.03 days ties to lunar draconic across the range 22.13 to 22.14:

    22.13 / 2 = 11.065
    22.14 / 2 = 11.07

    0.0745030006844627 = 27.212221 / 365.25

    0.0740047112086532 = axial(11.065,0.0745030006844627)
    0.0740049347670639 = axial(11.07,0.0745030006844627)

    27.0302207689606 = 0.0740047112086532 * 365.25
    27.0303024236701 = 0.0740049347670639 * 365.25

    27.03 day ref:
    Neugebauer, Ruzmaikin, Feynman, & Vaughan (2000). The solar magnetic field and the solar wind – existence of preferred longitudes. JGR.

  189. Paul Vaughan says:

    Crude comparative hindsight ignoring bundle$struck$”sure!” D-tale$….

    7920^2=62726400
    62726400/64000=980.1
    980.1/2=490.05
    490.05 ~= 22.135 * 22.139
    490.05*3=1470.15
    1470.15*25=36753.75
    36753.75*2=73507.5

    73500/2=36750
    36750/25=1470
    1470/3=490
    490 ~= 22.135 * 22.137
    490*2=980
    980*64000=62720000
    √62720000=7919.59594928933

    7920.40403, 62732800, 980.2, 490.1 ~= 22.137 * 22.139, 1470.3, 36757.5, 73515

    …but the relationships are 11inear in frequency — not period — and:
    0.000__%errors compared with
    0.00000___%errors guided diagnostics past this crude memorrery aid for “980 & 1470 lovers” too strained buy D-tale$.

    Crudely indulging D-taste for mmm = misunderstanding, misinterpretation, and misrepresentation.
    Yes (sarc) ‘b!g oi11’ pays for luck D-own calculation $0[]well (/sarc). Fact: check$, U Never.ca.me.

  190. Paul Vaughan says:

    Estimating JEV sidereal from J & S sidereal & tropical, 70^2, & M11:

    22.1370383309738 = 7920/√(2*64000.2003306175)
    22.1348489655186 = 490 / 22.1370383309738
    22.1392281295736 = beat(22.1348489655186,11.0685191654869)
    22.1392314983837 = Seidelmann (1992) sidereal JEV period
    -0.000015216473 = % error

    Comparing JEV sidereal estimate from J & S sidereal & tropical, 70^2, & M11 with JEV estimate from 9 & J & S sidereal (without tropical) :

    22.1392294366592
    22.1392281295736
    0.000005903935 = % error

  191. Paul Vaughan says:
    e=(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^1)^(0/1)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^2)^(2/2)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^3)^(3/3)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^4)^(2/4)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^5)^(5/5)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^6)^(1/6)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^7)^(7/7)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^8)^(4/8)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^9)^(6/9)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^10)^(3/10)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^11)^(11/11)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^12)^(4/12)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^13)^(13/13)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^14)^(5/14)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^15)^(7/15)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^16)^(8/16)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^17)^(17/17)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^18)^(6/18)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^19)^(19/19)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^20)^(8/20)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^21)^(11/21)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^22)^(9/22)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^23)^(23/23)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^24)^(8/24)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^25)^(20/25)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^26)^(11/26)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^27)^(18/27)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^28)^(12/28)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^29)^(29/29)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^30)^(9/30)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^31)^(31/31)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^32)^(16/32)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^33)^(19/33)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^34)^(15/34)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^35)^(23/35)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^36)^(12/36)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^37)^(37/37)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^38)^(17/38)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^39)^(23/39)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^40)^(16/40)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^41)^(41/41)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^42)^(13/42)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^43)^(43/43)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^44)^(20/44)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^45)^(24/45)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^46)^(21/46)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^47)^(47/47)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^48)^(16/48)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^49)^(42/49)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^50)^(20/50)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^51)^(31/51)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^52)^(24/52)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^53)^(53/53)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^54)^(18/54)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^55)^(39/55)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^56)^(24/56)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^57)^(35/57)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^58)^(27/58)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^59)^(59/59)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^60)^(16/60)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^61)^(61/61)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^62)^(29/62)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^63)^(36/63)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^64)^(32/64)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^65)^(47/65)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^66)^(21/66)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^67)^(67/67)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^68)^(32/68)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^69)^(43/69)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^70)^(25/70)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^71)^(71/71)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^72)^(24/72)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^73)^(73/73)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^74)^(35/74)/(1-(1/2/cos(s(71^2)/s(59^2)/s(47^2)*8*π))^75)^(40/75)/...
  192. Paul Vaughan says:

    The Perfect Garden

    The 3 lowest perfect numbers:
    s(6)=6; s(28)=28; s(496)=496

    The perfect key to the dimension of monster group minimal faithful representation:
    59=s((496+28+6)/2)
    59=s((s(496)+s(28)+s(6))/2)
    59=s(836/2-67-43-19-11-7-3-2-1) —————— note well: all non-163 Heegner nos!uphere

    The smallest weird untouchable number:
    836=496+28+6+2*(1+2+3+7+11+19+43+67) ————— again hear:
    836=s(496)+s(28)+s(6)+2*(1+2+3+7+11+19+43+67) —- ALL non-163 Heegner numbers appear
    836=(196883-59^3)/59+980

    [WHEEL CLOCKS] Don’s Guard [DAV]oscar[ID]UN D-[COV]air sov. Lo$Scar[e] (Bridge Fare[$]Road)
    980=836+144
    980=836+12^2
    980=836+(59^3-196883)/59
    980=2*(496-6)=2*(s(496)-s(6))
    490=496-6=s(496)-s(6) —————————– peerfactlieUNdarestunDemoCrazy
    =
    B-lock WHO’11 sun?
    B-lock Hale’s: UN!
    Washuawei D-Reign…………. — Sound Guard don[e]
    =
    1470=3*(s(496)-s(6))
    73500=75*(836+(59^3-196883)/59)
    73500=75*(836+(s(836/2-67-43-19-11-7-3-2-1)^3-196883)/s(836/2-67-43-19-11-7-3-2-1))

    260=(836-163-67-43-19-11-7-3-2-1)/2 —– my!my!my!yen!more!!beer!mmm…

    196883 = 47 * 59 * 71
    196883 = (59-12)*59*(59+12)
    196883 = 59^3 – 59*12^2
    196883 = 59^3 – 59*144
    196883 = 59^3 – 59*(980-836)

    “Times are gone for honest men” — Soundgarden

  193. Paul Vaughan says:

    US Route 22: West Turn Math

    Except shown ally sporadic McKey, He ignore side real true pick calls at turn Joe putter.

    64000.26272578 = 2*100/(1/(19+43+67+163)-s(836/2-67-43-19-11-7-3-2-1)/196883)

    where:
    100 = sum of Heegner nos. each mod 24
    24 = 1+2+3+7+11 = sum of LOW west [turn up at math] Heegner nos. [AD:UK$UN]
    292 = 19+43+67+163 = some of HIGH est Heegner know$[up eerie ‘Or’ ski11$m[y]th]
    836 = smallest weird UNtouchable number
    196883 = D-mention of mmminimmmummm faith full-monde star-grip Rep. Pres. sent Ace Sun
    59 = s(836/2-67-43-19-11-7-3-2-1) where s(RI[!]NO[!]) D-note$(“ally”[!]caught[!]sum) :

    1+2+3+7+11; 19+43+67+163

  194. Paul Vaughan says:

    AB out 111’s CRude$t “Irreducible” “Faithful” Point

    111.293375394322 = axial(490,12^2)
    111.292543528394 = harmean(164.791315640078,84.016845922161)
    0.000747458815 = % error

    φ(111) = 8^2 – 5^2 = 39

    1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8^2

    Split this sum into 2 categories:
    A. not prime:
    1 + 9 + 15 = 25 = 5^2
    B. prime:
    3 + 5 + 7 + 11 + 13 = 39 = 64 – 25 = 8^2 – 5^2 = 39

    13# = (8^2-5^2)*(7920/10-22) = φ(111)*(7920/10-22) = 39*(7920/10-22) = 13*11*7*5*3*2 = 30030

    “[…] 11 points gives a complex irreducible representation in 10 dimensions. This is the smallest possible dimension of a faithful complex representation […]”

    Piece Fully

    This is the “m[y]th or math?” series and “no. free speech” luminaries miss the point of hope and peace 836/209=$128000/$32000 a11.
    $sea weather UK can figure IT out “peace fully” with “0.000000000000 = % error” “green a11 lie$” (already shown).

  195. Paul Vaughan says:

    PRO Moon No.[39]Morale

    “With a word [11] Can. ‘get ‘ what sh[e!]k aim 836/209=$128000/$32000″ -Led 22plan

    196883 = s(836/2-67-43-19-11-7-3-2-1)^3-s(836/2-67-43-19-11-7-3-2-1)*(980-836)

    There’s no. room left in the dug-out grean (not a typo) music bees ID “sentimentale”. Eye sun how ear Norton honey moon here’s $64000 quest yen.

    =
    President Dwight […] did not want to be disturbed while the show was on […] Foghorn […] “Fox-Terror” […] spoof […] “reaching the first plateau” […] Scandal and cancellation […] fictitious $99,000 Answer […] Art Carney […] who identified three times in a man’s life when he wants to be alone, with the third being “when he’s in the isolation booth of The $64,000 Question” […] answering the 11th question […] contestant had been given answers in advance […] the so-called “Original 39”
    =

  196. Paul Vaughan says:

    Review, with 3 Suggestions

    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    This differs from Seidelmann’s (1992) Jupiter-Earth-Venus cycle period by 3.01679387133036 seconds per century. Above I showed the limit. I urge readers to consider this: why does 11 give the sharpest insight here?

    22.1348478401313 = (245/(φ^22+1/11)^(e/11+1/22)+√((245/(φ^22+1/11)^(e/11+1/22))^2+980))/2

    111789.639177999 = beat(22.1392315068494,22.1348478401313)

    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)
    This differs from Seidelmann’s value by 1.493352581686 seconds per century.
    Suggestion: Study Schneider’s delightful proof and strive to understand why replacing φ (which gives e) with 240 which gives Earth’s sidereal orbital period so precisely.

    1.00002637966847 = beat(111789.639177999,1.00001743390371)
    This is off from the terrestrial anomalistic year period by 17.8431319091906 seconds per century.

    Going through this the other way, starting with terrestrial anomalistic & sidereal years and Seidelmann’s JEV to arrive at 128000:

    111759.01908408 = beat(1.00002638193018,1.00001743371442)
    22.1348466308557 = axial(111759.01908408,22.1392314983837)
    22.1370388474829 = harmean(22.1392314983837,22.1348466308557)
    128000.394688135 = (7920 / 22.1370388474829)^2
    128000.400661193
    -0.000004666437 = % error

    1.00002638193018 is a placeholder value not yet scrutinized.
    Others might begin there exploring further refinement.

  197. Paul Vaughan says:

    Nominal Simplicity

    378 = monster factors sum
    496 = s(496) = 378+2*59
    490 = 496-6 = s(496)-s(6) — difference of perfect numbers

    These notes are not exhaustive, but they should be enough to clue some folks in:
    980 = 2*(496-6) = 2*(s(496)-s(6))
    980 = 836 + 144 = 836 + 12^2 = 836 + (59^2-196883/59) = 836 + (163-19)
    836 = smallest weird untouchable no.
    504 = 2*836-4*(163+67+43+19)
    418 = 836 / 2
    356 = 2 * 178
    209 = 836 / 4
    178 = baby monster factors sum
    100 = sum of Heegner nos. each mod 24
    50 = sum of Heegner nos. each mod 19

    14C & 10Be periods give us a lesson on modular features of nonlinear stability.
    In the days before Seidelmann’s (1992) precision, “2400 year” probably took memorable root from nominal integer origins:

    2400.12903225806 = beat(418,356)
    1200.06451612903 = beat(209,178)

    The nominal cases look quite simple with earned hindsight.

  198. Paul Vaughan says:

    Diagnostics

    More diagnostics on 70^2 and M11 appearance in solar system stability. This makes pretty clear the way Jupiter, Saturn and Venus (which is coupled with Earth) shape Earth’s anomalistic year.

    From Laskar
    183829.787234043 = 1 / s_2 ——- Venus node
    173913 = 1 / g_2 ———————- Venus perihelion
    68753.3156498674 = 1 / s_3 ——- Earth node

    178733.967728589 = harmean(183829.787234043,173913.043478261)
    111733.770152599 = beat(178733.967728589,68753.3156498674)

    With Seidelmann (1992) sidereal Earth year:
    1.00002638395227 = beat(111733.770152599,1.00001743371442)
    1.00002638193018 = 365.259636 / 365.25 ——– Earth anomalistic year
    0.000000202203 = % error (absolute: 15.9525968265692 seconds per century)

    With Seidelmann (1992) sidereal Jupiter-Earth-Venus (JEV) :
    22.1348456401858 = axial(111733.770152599,22.1392314983837)
    22.1370383520498 = harmean(22.1392314983837,22.1348456401858)

    489.999926853495 = 22.1348456401858 * 22.1370383520498
    490
    -0.000014927858 = % error

    128000.400417504 = (7920 / 22.1370383520498)^2
    128000.400661193 = 1 / ( –Jside/2 + Sside/2 + Jtrop/2 – Strop/2 )
    -0.000000190381 = % error (absolute: -15.0199398937022 seconds per century)

    It’s a beautiful sight.

    Next, J & S sidereal & tropical with JEV, all from Seidelmann (1992) :
    22.1370383309774 = 7920 / √128000.400661193
    11.0685191654887 = 22.1370383309774 / 2
    22.1348455980494 = beat(22.1392314983837,11.0685191654887)
    489.999925454285 = 22.1348455980494 * 22.1370383309774
    490
    -0.000015213411 = % error

    111732.696485953 = beat(22.1392314983837,22.1348455980494)
    1.00002638403827 = beat(111732.696485953,1.00001743371442)
    1.00002638193018 = 365.259636 / 365.25 ——– Earth anomalistic year
    0.000000210803 = % error (absolute: 16.6311136296105 seconds per century)

    111733.770152599
    111732.696485953
    0.000960924313 = % error

    68752.9091228114 = axial(178733.967728589,111732.696485953)
    68753.3156498674 = 1 / s_3 ———- Earth node
    -0.000591283565 = % error

    Anyone taking this further, I suggest starting by scrutinizing this value, because I haven’t yet done so:

    1.00002638193018 = 365.259636 / 365.25 ——— terrestrial anomalistic year

  199. Paul Vaughan says:

    Wish!own Tune Found Door Well

    59 = average(+496, -2-3-5-7-11-13-17-19-23-29-31-41-47-59-71)
    31 = average(+2+3+5+7+11+13+17+19+23+29+31+41+47+59+71, -1-2-3-7-11-19-43-67-163)

    61.0461538461538 = harmean(1984, 31)
    61.0464822565173 = slip(29.4474984673838,11.8626151546089)
    -0.000537967711 = % error

  200. Paul Vaughan says:

    Tee Ignore

    Look past what tops 1984 “perfect” try gone:

    71 = 42+21+8
    71 = φφ/(J+S)+1/(J-S)+1/S-1/(J+S)+1/(J-S)-1/J ———————- frequencies
    71 = φφ/(J+S)-1/(J+S)+1/(J-S)+1/(J-S)+1/S-1/J
    71 = φ/(J+S)+2/(J-S)+1/S-1/J
    71 = φ/(1/j+1/s)+2/(1/j-1/s)+s-j
    71 = 1.61803398874989/(1/j+1/s)+2/(1/j-1/s)+s-j ———————- periods
    s = 29.44889869193547 and j = 11.863057204318362
    s = 29.44889869193547 and j = 11.863057204318362

  201. Paul Vaughan says:

    The Miss Sun Link

    216 = 378-Φ(163)
    216 = (2+3+5+7+11+13+17+19+23+29+31+41+47+59+71)-Φ(163)

    271 = 163+Φ(378)
    271 = 163+Φ(2+3+5+7+11+13+17+19+23+29+31+41+47+59+71)

    2,3,5,7,11,13,17,19,23,29,31,41,47,59,71

    Leaving no ambiguity:
    Standby for peaceful resolution.

  202. Paul Vaughan says:

    Clarity

    In memory of Srinivasa Ramanujan, whose precious life ended at a mysteriously young age:

    58 = 19-MOD(19,24)+43-MOD(43,24)+67-MOD(67,24)+163-MOD(163,24)-10-13-18-22-37-58

    keywords: 104, 744, Heegner numbers

  203. Paul Vaughan says:

    Quick Review of Ramanujan’s 104 with Plato

    104.212132286568 = ⌊(e^√10π)^(1/2)⌉^2 – e^√10π
    -103.947369666712 = ⌊(e^√13π)^(1/2)⌉^2 – e^√13π
    104.007114381762 = ⌊(e^√18π)^(1/2)⌉^2 – e^√18π
    104.001742574386 = ⌊(e^√22π)^(1/2)⌉^2 – e^√22π
    -103.999977946281 = ⌊(e^√37π)^(1/2)⌉^2 – e^√37π
    = round(exp(37^(1/2)*pi())^(1/2),0)^2-exp(37^(1/2)*pi())
    104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π

    Plato’s number = 216 balances:
    58 = 216-10-13-18-22-37-58

  204. Paul Vaughan says:

    Heegner 744 Review with Plato

    316 = 1+2+3+7+11+19+43+67+163

    100 = MOD(1,1+2+3+7+11)+MOD(2,1+2+3+7+11)+MOD(3,1+2+3+7+11)+MOD(7,1+2+3+7+11)+MOD(11,1+2+3+7+11)+MOD(19,1+2+3+7+11)+MOD(43,1+2+3+7+11)+MOD(67,1+2+3+7+11)+MOD(163,1+2+3+7+11)

    216 = (1+2+3+7+11+19+43+67+163) – ( MOD(1,1+2+3+7+11)+MOD(2,1+2+3+7+11)+MOD(3,1+2+3+7+11)+MOD(7,1+2+3+7+11)+MOD(11,1+2+3+7+11)+MOD(19,1+2+3+7+11)+MOD(43,1+2+3+7+11)+MOD(67,1+2+3+7+11)+MOD(163,1+2+3+7+11) )

    216 = (19+43+67+163) – ( MOD(19,1+2+3+7+11)+MOD(43,1+2+3+7+11)+MOD(67,1+2+3+7+11)+MOD(163,1+2+3+7+11) )

    -743.777680155239 = round(exp(19^(1/2)*pi())^(1/3),0)^3-exp(19^(1/2)*pi())
    -743.999775171279 = round(exp(43^(1/2)*pi())^(1/3),0)^3-exp(43^(1/2)*pi())
    -743.999816894531 = round(exp(67^(1/2)*pi())^(1/3),0)^3-exp(67^(1/2)*pi())

    744 ~= ⌊ ⌊(e^√163π)^(1/3)⌉^3 – e^√163π ⌉ —- not all calculators can handle this calculation

  205. oldmanK says:

    PV’s ‘number salad’ – Plato’s number – this time has cherries as a main ingredient. TY Paul.

    I had no idea what lurked behind it so I looked it up. This here ref is important.
    https://www.academia.edu/243917/Four_Mathematical_Texts_from_the_Temple_Library_of_Nippur

    Apparently Plato seemed influenced by the sexagesimal counting of the Babylonians. Quote “–what proved to be a very successful numerological-mythological manipulation of a much older Babylonian story”.
    And: “It goes without saying that the number 3600 rests upon the Babylonian sexagesimal system.”

    Then “— in the life of the universe, in which the world waxes and wanes alternately.” This seems to hark back to the Eddy cycle. Plato was not referring to the last 2kyr but to much earlier times.

    Further “expressed in years, 360 days counted per year,” Why? And when “Seven, eleven and thirteen are not divisors of 12,960,000, and to this day are still regarded as unlucky numbers.” Seven times thirteen equals ninty one (7*13=91) is the number of days per season to make a year of 364 days, with 7 days per week. There is other evidence that the Akkadian world had knowledge of -and links to- the megalithic calendar (where the best divisor scale for the quadrant was of 90 divisions).

    Further down “We know from Berossus, writing in the third century BC, that a period of 36,000 years, is called the ‘Great Platonic Year’, in early astronomical treatises, and that this was the duration of a Babylonian cycle.” Both Plato and Berossus had tales about changes and disturbances of the heavens that were permanent.

    Additional “On the contrary, in Plato’s words: ‘Whenever our guardians promote marriages inopportunely, the offsprings cannot be well provided.’. Since working on this subject I have had occasion to note from ancient graves, and mediaeval dress of both sailors and aristocracy, periodic change in human stature, in line with Eddy cycle.

    However there is quite a long time between 4th millennium Akkadians and Plato. Did Plato get the story and the numbers right?

  206. Paul Vaughan says:

    oldmanK, herd insight avalanches Norton called “the voice of God”.

    typo above
    744 ~= ⌊ ⌊(e^√163π)^(1/3)⌉^3 – e^√163π ⌉
    should read
    744 ~= | ⌊(e^√163π)^(1/3)⌉^3 – e^√163π |

    May peace be with you.

  207. Paul Vaughan says:

    Jupiter-Saturn Axially 323, not 322

    One narrative contends solar system order’s based on Lucas numbers.

    1.62053454741122 = √(100/(1/11.8626151546089+1/29.4474984673838)/322)
    1.61803398874989 = φ
    0.154543024356 = % error

    The error grows absolutely with scale.
    An accurate Lucas model relies on hierarchically nested Lucas & Fibonacci corrections.
    Found: neat fractal correction = delight fully unnecessary.

    Note nearby. Comparison sharpens further.

    1.61650633853492 = √(1/(1/11.8626151546089+1/29.4474984673838)/(1+√5))
    1.61803398874989 = φ
    -0.094413975578 = % error

    1.6180240353733 = √(100/(1/11.8626151546089+1/29.4474984673838)/323)
    1.61803398874989 = φ
    -0.000615152504 = % error

    Key lessun: 323 = 196883-196560 fits ~250 times better than 322.

    Ramanujan supposedly had nearly-incomprehensible intuition about numbers (consciously distinct from ‘most logic AI’).
    ‘Correct’ exploration scales (but not politically) this shape: 262537412640768744.

    322.996026177168 = axial(26253741.2640769,323) = 1/(1/323+1/26253741.2640769)

    1.61803398862436 = √(100*(1/323+1/26253741.2640769)/(1/11.8626151546089+1/29.4474984673838))
    1.61803398874989 = φ
    -0.000000007759 = % error

    Sporadic simple group math historians may find AImost-lukeUS numbers act sea ally enlightening joviun-fractal hindsight.

    8.43006944377466 = φφ*322/100 __________________________ -0.308371015889 = % error
    8.47213595499958 = φφ*(1+√5) = 2*φ^3 _____________________ 0.189095708159 = % error
    8.45624978366216 = φφ*323/100 ___________________________ 0.001230316360 = % error
    8.45614574762977 = φφ/(1/323+1/26253741.2640769)/100 ______ 0.000000015517 = % error
    8.4561457463176 = axial(Jupiter period, Saturn period)

    “if ITkeepsunreignunloveease going to B___” — Led 22plan

    May peace be with you.

  208. Paul Vaughan says:

    Van Hale UN

    “but then my homework was never quite like this”

    5 is to 12 as
    10 is to 24.

    ZZ(12)ignore(34)republic(56)’cant’op“11nos.how22US$mmm” ($seenITorwell).

    big dipper side wise quest yen-mark
    5 = average(163,-67-43-19-11-7-3-2-1)

    play dough nos. bog deeper
    12 = √(3456+5*5-47*71)
    3456 = (59-5)*(59+5)

    morals 12-G no. rome mmmore SIM$UN
    12 = 71-59 = 59-47
    3^3 = 27
    4^3 = 64
    5^3 = 125; 125+64+27 = 216; 216000 = 125*64*27 —– “memoria” nirvana
    6^3 = 216

    perfect play “doh!”
    12/2 = 6 = s(6)
    34 = s(6)+s(28)
    56/2 = 28 = s(28)

    71-47 = 1+2+3+7+11
    orion can reach auriga
    (71+5)-(47-5) = 28+6 = 163-67-43-19

    May peace be with you. Response[ability] “and also with you.”

    “Mathematics is the queen of the sciences—and number theory is the queen of mathematics.” — Gauss

  209. tallbloke says:

    Thanks for drawing our attention back to this post Paul. I don’t understand what your comment is driving at, but Oldbrew and I were just discussing longer term gas giant cycles that appear to be linked to Milankovitch and other important EOP periodicities. More soon.

  210. Paul Vaughan says:

    Optimal Nonlinear Seas Time

    TB surely seas each number has to be understood in the context of every other number…
    …and maybe only God nos. the full depth available to sea.

    Since ‘time limits’ supply ‘no. infinite tee’, Pareto Principled learning can balance insight…
    …e.g. net zero:

    “The existence of such an integral vector of Lorentzian norm zero relies on the fact that 1^2 + 2^2 + … + 24^2 is a perfect square (in fact 70^2); the number 24 is the only integer bigger than 1 with this property. This was conjectured by Édouard Lucas […]”

    Tip for those awakening to the nonlinear nature of our number system:
    73 is the lowest prime congruent to 1 mod 24; “24 is 5hemiperfect“.

  211. Paul Vaughan says:

    31 Lincoln

    If f(UN) awoke can “Sporadic and Exceptional” mystery US sly points to 42, 104, & 163.
    73 = average(42,104) = average(19,43,67,163)

    177 = 104+73 = 323-104-42
    177 = 288-111 = 47+59+71
    Φ(196883-196560) = Φ(323) = 288

    496 = 378+177-59
    496 = 378+288-111-59
    496 = 378+Φ(323)-111-59
    496 = 378+Φ(Φ(1333))-111-59
    496 = 378+Φ(196883-196560)-111-59

    496 = 4^2 * ( 5^2 + s(5^2) ) = 16*(25+6) = 16*((31))

    490 = 4^2 * ( 5^2 + s(5^2) ) – s(5^2) = 16*25 + (16-1)*6
    490 = s(4^2*(5^2 + s(5^2))) – s(5^2) = s(496)-s(6) = 496-6

    490 = 16*25 + 15*6
    490 = 16*25 + 15*6 ————– ((31)) = 16+15
    490 = 16*25 + 15*6 ————– ((31)) = 25+6

    378 = 2+3+5+7+11+13+17+19+23+29+((31))+41+47+59+71 ——- sum of supersingular primes
    316 = 2+3+5+7+11+13+17+19+23+29-((31))+41+47+59+71

    A simple polarization isolated strict equality:

    -316 = -1-2-3-7-11-19-43-67-163 —————————————————— Heegner nos.
    ((31)) = average(-316,378) = 16+15 = 25+6 = 496/16

    Summary:
    31 = 25 + s(25)
    31 = ‘kingpin link UN’ 163 to supersingular primes (monster ‘group factors’) of 70^2 seen threw perfect no$. 496 & 6.
    31 is the only sporadic simple group prime dividing perfect number 496.
    31 = average(-42,104) ————————- completing “Sporadic and Exceptional” in-the-loop

    Together let US find peace, prosperity, and tranquility shared equally by all.

  212. Paul Vaughan says:

    Ramanujan’s Perfect Links to Heegner, Monster, & Baby

    6=s(6); 28=s(28); 496=s(496) — the 3 lowest perfect numbers

    323 = 196883-196560
    323 = average(496,28+6+2*58)
    323 = average(496,28+6+316-378+178)

    19 = (496-28-316)/8 = 163 mod 24 = 67 mod 24 = 43 mod 24 = 19 mod 24
    38 = (496-28-316)/4
    76 = (496-28-316)/2 = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24
    152 = 496-28-316
    304 = 2*(496-28-316)
    608 = 4*(496-28-316) —— s(608) = s(652) = s(4*163) = s(496) = 496
    1216 = 8*(496-28-316)
    2432 = 16*(496-28-316)

    4900 = 70^2
    490 = 496-6
    49 = 73-24 = 73-11-7-3-2-1

    73 = average(496-6,47*71-59*59-378+178)
    73 = average(496-6,-28-316) = average(19,43,67,163)
    146 = 496-6-28-316 = average(19+43,67+163)
    292 = 2*(496-6-28-316) = 19+43+67+163 = 316-11-7-3-2-1

    104 = d(4,1/2,58) = -d(2,1/2,37)

    216 = average(316,2*58) = Plato’s Number
    360 = 323 + s(323) = 323 + 37 = 1 full circle in degrees

  213. Paul Vaughan says:

    ‘Sort of’ Organizing 100, 70, & 260 from Heegner, Monster, & Baby

    100 = sum of Heegner numbers each mod 24
    100 = average(378,-178) = average(316,-2*58)

    316 = Heegner number sum
    216 = Plato’s Number = average(316,2*58)
    116 = 2*58

    378 = supersingular prime sum = unique monster factor sum
    178 = unique baby monster factor sum

    200 = 29+41+59+71 — sum of monster factors that are not baby monster factors

    With balanced partitioning:
    100 = 29+71
    100 = 59+41

    It’s not as effortlessly recognizable with Mayan symbolic partititioning:
    70 = 29+41 = 378-178-71-59 = 316-2*58-71-59
    130 = 59+71 = 378-178-41-29 = 316-2*58-41-29
    260 = 59+71 + 378-178-41-29 = 316-2*58-41-29 + 59+71

    Monster order:
    196883 = 47*59*71 = (59-12)*59*(59+12)
    dividve by 59:
    3337 = 47*71 = (59-12)*(59+12) = 59*59-12*12
    rearrange pieces:
    144 = 12*12 = 59*59-47*71

    Summary:
    In sober second: thought.
    Ancients knew “monstrous moonshine” but maybe they simply didn’t call it that.
    Imagine there: why’s reaction? “A future tribe calls this mathematical structure ‘baby monster’.”
    Insider trade secrets well-protected. Buy awakeUNspeakable name only.
    Before creative labeling: ‘just math’. Knew regulation names “The Party” of [re]discovery.

    I’ll share a few more examples. That may be enough.

  214. Paul Vaughan says:

    WHO’s Home M[mmm]ore SIM$UN Play “Doh?”

    Comm. mist tree lag rains point sh!ape.

    216 = 489426/43/7/3/2-2-3-5-7-11-13-17-19-23-29-31-41-47-59-71+(196883-196560)
    216 = (163+Φ(2+3+5+7+11+13+17+19+23+29+31+41+47+59+71))-2-3-5-7-11-13-17-19-23-29-31-41-47-59-71+(196883-196560)

    “Apart from all other reasons, the parameters of the geoid depend on the distribution of water over the planetary surface.” – Nikolay Sidorenkov

    52 = (196883-196560)-(163+Φ(2+3+5+7+11+13+17+19+23+29+31+41+47+59+71))
    52 = 323-271 = (196883-196560) – 489426/43/7/3/2

    37 = average(-55, 129) = average(-(28^2-27^2), 19+43+67) = average(-(5^2+4^2+3^2+2^2+1^2), 19+43+67)
    37 = s(196883-196560)
    52 = average(⌊(e^√37π)^(1/1)⌉^1-e^√37π, e^√37π-⌊(e^√37π)^(1/2)⌉^2)

    58 = average(1+2+3+7+11+19+43+67,-s(196883-196560))
    52 = average(e^√58π-⌊(e^√58π)^(1/1)⌉^1, ⌊(e^√58π)^(1/k)⌉^k-e^√58π) for k=2,4

    “Mathematics is the queen of the sciences—and number theory is the queen of mathematics.” — Gauss

    180 = 496-1-2-3-7-11-19-43-67-163 = average((196883-196560),s(196883-196560))
    360 = 496-1-2-3-7-11-19-43-67-163 + average( (196883-196560),s(196883-196560))

    360 Degrees

    153 = (52 + 52) + 70^2 / 2^2 / 5^2 = 104+(25^2-24^2)
    153 = 37+2*58
    153 = 111+42 = 104+49

    153 = mod(496-163+average((196883-196560),s(196883-196560)),360)
    153 = 496-163+average((196883-196560),37)-360
    153 = 1+2+3+7+11+19+43+67
    153 = 5^3 + 3^3 + 1^3 ——————————————————– sci11UNs
    153 = 496-7^3 ———————————————————– comm. UN IC case sun

    153 = mod(average(836,496)-153,360)
    153 = average(836,496)-153-360
    323 = mod(836-153,360)
    323 = 836-153-360

    496 = 7^3 + 5^3 + 3^3 + 1^3 —————————— perfect borg cube primes (method of loci)
    496 = 7^3 + 1+2+3+7+11+19+43+67
    496 = (5^3 + 3^3 + 1^3 + 163) + 180 = 316 + 180
    496 = 7^3 + mod(496-163+average((196883-196560),s(196883-196560)),360)

    “Win tour” warm up exercise:
    271 = 216+(1^2+2^2+3^2+4^2+5^2)
    378 = 323+(28^2-27^2) = 323+(27+28)
    100 = average(-178,378)

  215. Paul Vaughan says:

    Knew Don wins ‘heaven too’ weight?

    3456 J

    in very hint half not iced?

    “the sweet test: tap, pull on the tree” – David Wilcox

  216. Paul Vaughan says:

    Everyone’s “obviously” aware?

    59 = s((496+28+6)/2)=s((s(496)+s(28)+s(6))/2) —- 3 perfect numbers
    59 = s(836/2-67-43-19-11-7-3-2-1)
    59 = s(836/2-67-43-19-24)
    59 = s(163+67+59-24)
    59 = s(163+43+59+0)
    59 = s(163+19+59+24)
    59 = s(260-19+24)
    59 = s(490-163-43-19)
    59 = s(490-163-67-19+24)

    59 = average(496/2,-260/2)
    59 = average(71,47)
    260 = average(496,71-47)
    378 = average(496,260)

    no. doubt…

  217. Paul Vaughan says:

    Earth-Venus Modular Resonance

    Sharp readers will notice a key detail left out (already covered above).

    _
    Note a divergence crystallizing at level 5.

    sidereal slip hierarchy review
    1.59868955949705 = beat(1.00001743371442,0.615197263396975)
    2.67043906295843 = slip(1.00001743371442,0.615197263396975)
    8.10187610587681 = slip(2.67043906295843,1.59868955949705)
    238.911497097738 = slip(8.10187610587681,2.67043906295843)
    489.155247011449 = slip(238.911497097738,8.10187610587681)

    tropical slip hierarchy
    1.59868953279706 = beat(0.99997862,0.61518257)
    2.67016237493693 = slip(0.99997862,0.61518257)
    8.09678584788177 = slip(2.67016237493693,1.59868953279706)
    250.522048989645 = slip(8.09678584788177,2.67016237493693)
    4240.7928948521 = slip(250.522048989645,8.09678584788177)

    sidereal with tropical basic review
    25757.05496809 = beat(0.615197263396975,0.61518257)
    25760.5207691867 = harmean(25763.987503107,25757.05496809)
    25763.987503107 = beat(1.00001743371442,0.99997862)

    hierarchical extension to modular resonance drift
    25770.9237709108 = slip(2.67043906295843,2.67016237493693)
    25774.3933055453 = 2 * slip(8.10187610587681,8.09678584788177)
    25775.0873245853 = 5 * slip(250.522048989645,238.911497097738)
    Queens of the Stone Age — Live in London 2005 16278263789…
    25672.6551628575 = 2 * slip(4240.7928948521,489.155247011449)

    Note the reset after drifting up from ~25760. For comparison:

    25259.6956047047 = beat(164.791315640078,163.7232045)
    25679.8527338811 = harmean(26114.2236547808,25259.6956047047)
    25685 = Laskar’s p
    26114.2236547808 = beat(84.016845922161,83.74740682)

    Interpretation: Jovian responsibility for EV tropical offset from sidereal.

    Probing around the Seidelmann (1992) typo since December 2020 has brought many things swiftly and effortlessly into focus. My estimate based on new scope: it would take at least 70 years (and maybe several lifetimes) to thoroughly explore and report everything noteworthy — hence the sample reporting.

  218. Paul Vaughan says:

    UN Faithful Lyons Rep Reviews Baby Monster Aliquot

    ‘Plato’s backcountry treasure map’ won bright young clueseaSAMsun.
    Homer nos. Plato UN D-air$stand$’better’ then hymn…

    216 = 2*φ(Φ(163)) = 378-Φ(163) = ⌊harmean(4270,111)⌉

    437 = average(378,496)
    4370 = 2*5*average(378,496)
    4370 = dimension of minimum faithful representation of baby monster group
    4270 = s(4370)
    111 = dimension of minimum faithful representation of Lyons group.

    Nominal Level 1 Review

    4270 ~= beat(Uranus period, Neptune period / 2) = 1/(2N-U)
    111 ~= harmean(Uranus period, Neptune period) = 1/(U/2+N/2) where U & N are frequencies
    Neptune period ~= 3*axial(4270,111/2)
    Uranus period ~= 3*beat(4270,111/4)

    Nominal Level 2 Review

    144 = 59*59-47*71

    84.0174899796991 = 3/(4/144+4/490-1/4270)
    164.792485578641 = 3/(2/144+2/490+1/4270)
    171.407635533372 = 3/(2/144+2/490-2/4270)
    55.6466876971609 = 1/(2/144+2/490+0/4270)
    111.293375394322 = 1/(1/144+1/490+0/4270)

    111.293375394322 = axial(490,144)
    111.292543528394 = 2/(U+N) = harmonic mean = 2*axial period
    0.000747458815 = % error

    164.792485578641 = 3 * axial(4270,axial(490,144)/2)
    164.791315640078 = 1/N
    0.000709951588 = % error

    84.0174899796991 = 3 * beat(4270,axial(4270,axial(490,144)/4))
    84.016845922161 = 1/U
    0.000766581429 = % error

    171.407635533372 = 3 * beat(axial(4270,axial(490,144)/2),beat(4270,axial(4270,axial(490,144)/4)))
    171.406220601552 = 1/(U-N)
    0.000825484522 = % error

    Level 3 went past nominal (already covered) – but not with Plato.
    Thus left exercises 4D-elect ably rural state sov. US.

    Urban ‘fact’-check industry’s enslaved buy-master$sov. D-bait. Summary of their 489426 page report: Dem claim 2+2=5. So-called ‘trump supporters’ countered with 2+2=3. Link UN PR Joe actors then left no. doubt with their assurance 2+2=1 but not when 2+2=7.

    Link UN Creative Support PR Joe Act: Funding Feminine Rural Arts

    Harmoniously a rose peaceful rural math (minus thorn) uplifted past mmmost-outgoing type sov. ‘D-bait chasing’ “political dude-look”.

    IC Homer can’t tell a needle of truth from a haystack of lies and $0 needs a peaceful course well-away from master$ov. D-bait.

    May peace be with ‘you win the backcountry luke sea!’ – both during and after long-overdue recalibration for true piece weather left, right, home or abroad.

  219. Paul Vaughan says:

    The Free Won: Weather Left or Right ….and Especially Center

    Just in time, “the dawn of elect trick banjo tunes” peace by piece write in the middle of some “won freedom”. Note carefully Heegner, Plato, and Sylvester on Pythagorean steps offset at the base by 16 = 31-15.

    63 = 16+(24^2-23^2) = 16+(23+24) = 10+13+18+22 = 216-37-2*58
    163 = 63+(26^2-24^2) = 63+((24+25)+(25+26)) = 63+100

    “Open up Eur. rise….” — Queens of the Stone Age

    216 = 163+(27^2-26^2) = 163+(26+27)
    216 = 10+13+18+22+37+2*58 = sum of Heegner nos. minus sum of Heegner nos. each mod 24
    216 = 6^3 = 3^3+4^3+5^3 = 378-Φ(163)

    271 = 216+(28^2-27^2) = 216+(27+28) = 216+(1^2+2^2+3^2+4^2+5^2)
    271 = 489426/(2*3*7*43) —— this link gives a fantastic constraint
    271 = 163+Φ(378)

    100 = average(-178,378) ————– sum of B & M factors
    100 = 26^2-24^2 = sum of Heegner numbers each mod 24

    This natural framework accounts for Neptune & Uranus relations with Saturn & Jupiter. Here’s the coarse outline:

    4270 = s(4370) & 111 give first level nominal impression ——- B & Ly
    323 = 196883-196560 ———— M & Leech
    37 = s(196883-196560)
    52 = average(⌊(e^√37π)^(1/1)⌉^1, -⌊(e^√37π)^(1/2)⌉^2)
    271 = 323-52
    104 = round(exp(37^(1/2)*pi())^(1/1),0)^1-round(exp(37^(1/2)*pi())^(1/2),0)^2
    427 = 323+104 = 4270/5/2 = 316+111 ———– 316 is sum of Heegner nos.
    4270 = s(4370)

    Overview of more precise scaling initially quantized with Bollinger’s (1952) method:

    19 = x mod 24 for x = 19,43,67,163
    4370 = (23*5*2)*19
    11500 = (23*5)*(5*2)^2
    5.5 = 55 / (5*2)
    55 = 271-216 = 27+28 = 28^2-27^2 = 1^2+2^2+3^2+4^2+5^2
    55 = 163+Φ(378)-378+Φ(163) = 163+108-378+162
    111.5 = ⌊4/(U+N)⌉

    111.5/(5.5*(J+S)+1/11500) = 1/(U-N) = U & N beat period
    (-0.000000053635 = % error with Seidelmann (1992) periods

    1/(3*(J+S-1/2/3/7/43/271)/1806/2+S) = 4/(J+S+U+N) = jovian harmonic mean period
    0.000000000000 = % error with Seidelmann (1992) periods

    Using these J & S estimates to estimate U & N:

    84.016845845182
    84.016845922161
    -0.000000091623 = % error

    164.791315879816
    164.791315640078
    0.000000145480 = % error

    171.40622002178
    171.406220601552
    -0.000000338244 = % error

    55.6462717577651
    55.6462717641972
    -0.000000011559 = % error

    Won the weather ‘gets’ ‘ruff!’ ” — Dove ID Will C(-11)ox (in the ‘year of rome’, UN ‘no?morals’….)

    111.29254351553
    111.292543528394
    -0.000000011559 = % error

    Review: 1806 = 2*3*7*43 tied clearly to primary pseudoperfect numbers, Sylvester’s sequence, and greedy Egyptian fractions.

    Hindsight: 489426 = 271*(2*3*7*43) = 271*1806
    271’s origin is now peacefully clear.

    Wishing well-balanced peace, security, and freedom for “won human being” on Earth.
    May “the voice of God” sh!ape our intellectual capacity to negotiate a superior course for all.

  220. Paul Vaughan says:

    “Unity Honeymoon” Nears: CR Am. D-UN, a11 ice

    “Bizarre how they truncated the SAM cycle, acting as if it is trending and they are trying to control that

    72 = 1*((average(-71,163)^5+47^5)+(19^5+43^5+67^5))^(1/5)
    144 = 2*((average(-71,163)^5+47^5)+(19^5+43^5+67^5))^(1/5)
    216 = 3*((average(-71,163)^5+47^5)+(19^5+43^5+67^5))^(1/5)
    288 = 4*((average(-71,163)^5+47^5)+(19^5+43^5+67^5))^(1/5)
    360 = 5*((average(-71,163)^5+47^5)+(19^5+43^5+67^5))^(1/5)

    There are 163 reasons to exercise “sober second” thought: n=5.

    “[…] unless you have the “correct keys” to sequence the cells DNA – you will not be able to decode the information stored in it […]”

    “We’re very excited about this potential…the idea of being able to conceal information in plain sight […]”

  221. Paul Vaughan says:

    490 = Partition Function ( 19 )

    “Each step in the story is a work of art,’ Dyson says, “and the story as a whole is a sequence of episodes of rare beauty, a drama built out of nothing but numbers and imagination.”

    “The proof of Rademacher’s formula involves Ford circles, Farey sequences, modular symmetry and the Dedekind eta function.”

    partition function divisor sum recursion:
    a(n) = (1/n) * Sum_{k=0..n-1} sigma(n-k)*a(k), where sigma(k) is the sum of divisors of k (A000203).

  222. Paul Vaughan says:

    Since 1800: Lunisolar Sci11UNs 66 96 208

    Low NY, $0 lure Am. ‘big glue’ IT.

    “Whatchya doin’ Homer?” ——– Link
    ” ‘niffin’ [C]lue ” —————- UN PR Joe Act

    WHO lie weed’s lip sly CA11? “Comm. UN IC ace sun break D-own” — Led 22plan
    Past climate discourse ineffectively (with words ) failed to relate quantities now tersely D-find.

    Tropical
    0.0748024157783867 = axial(0.999978614647502,0.0808503463381246)
    18.6129709123853 = beat(0.0748024157783867,0.0745030006844627)
    8.84735293159855 = beat(0.0754402464065708,0.0748024157783867)
    179.333323110834 = slip(18.6129709123853,8.84735293159855)
    1879.26478947996 = slip(179.333323110834,5.99685290323073)
    491.132481334807 = slip(179.333323110834,16.8627856518082)
    IT’s “$sure!” DCoy Homer. “No. win to hole dem – know when to fool dem …when too’UK’awei…&winterUN…” – KR

    Anomalistic
    0.0748026830551271 = axial(1.00002638193018,0.0808503463381246)
    18.5964370693548 = beat(0.0748026830551271,0.0745030006844627)
    8.85109350901809 = beat(0.0754402464065708,0.0748026830551271)
    16.8899949413115 = beat(18.5964370693548,8.85109350901809)
    184.063510192393 = slip(18.5964370693548,8.85109350901809)
    600.241396282931 = slip(184.063510192393,5.99685290323073)
    1800.72418884878 = slip(184.063510192393,16.8899949413115)

    We11comm.22theJung11″ _UNsunRose$goG-ale meekCRews$soft dr. ink a maze UN for “IT’s human” experimentAI subjects.

    Sidereal
    0.0748026329881375 = axial(1.00001743371442,0.0808503463381246)
    18.5995319875902 = beat(0.0748026329881375,0.0745030006844627)
    8.85039257541183 = beat(0.0754402464065708,0.0748026329881375)
    16.8848913580487 = beat(18.5995319875902,8.85039257541183)
    183.158541510238 = slip(18.5995319875902,8.85039257541183)
    400.297181344069 = slip(183.158541510238,5.99685290323073)
    1200.8915440322 = slip(183.158541510238,16.8848913580487)

    Events can be exactly defined (strict mathematical equality) as deviations from modular central limits. A major multidisciplinary dove ID in superior climate communication thus “flies-free” (naively assume UN no. pole-IT talk Cal. interfere ruins).

    Anomalistic-Sidereal
    1201.70988258678 = beat(600.241396282931,400.297181344069)
    240.145602009621 = axial(600.241396282931,400.297181344069)
    5 ~= 5.00408865509281 = 1201.70988258678 / 240.145602009621
    3605.12964776031 = beat(1800.72418884878,1200.8915440322)
    720.436806028858 = axial(1800.72418884878,1200.8915440322)
    5 ~= 5.00408865509282 = 3605.12964776031 / 720.436806028858

    360 lovers no. 120’s 3-perfect: σ(120) = 360

    Can instantly “The Party mmmyen!” $sagely ‘wise dem’:
    0.0745030006844627 = 27.212221 / 365.25 ———- draconic
    0.0754402464065708 = 27.55455 / 365.25 ———– anomalistic
    0.0808503463381246 = 29.530589 / 365.25 ———- synodic

    Peacefully sea weather pieces really fit to gather 22 states no. win sol ID err IT.

    Parody in Atlantic ‘Cause zones’
    3.90828049899173 = slip(1.00002638193018,0.0754402464065708) ——— sea-hawk key puck
    Homer’s DCoy alias$sing “AMO!”
    66.0278743780447 = slip(22.1392314983837,3.90828049899173) ———– fict.IT.US$sly!lieUN$

    Less extreme missed link con project:
    1728 = (3^3+4^3+5^3)*2^3 = 6^3*2^3 = 216*8 = (2*3)^3+(2*4)^3+(2*5)^3 = (2*6)^3
    Just weight, eventually EU’11 sea peacefully-blue collar biweekly$secures$financial equal=IT of comm.pose!ITory: all numbers like $1728.

    Be[11]owe$EU$IC(AI)
    210.705073993676 = beat(96.1613372617316,66.0276945924851) …QBO’s[11]ips$like a11

    Response 2 north poll quest yen ABout flywheel winds$: “$suddenly ice-sea!” KTUNsta11

    Multidecadal NAO multidecadal “AMO” (really Hazzard US ‘comm. Bo-nation’ sov. 3-count tease). IT’s “snot on the net” rich herd land$ “UN mail!” IT tory (“doh!”) “river of climate” contro[11]vertsy’s profITable fact chuck king west turn book$$sail$$MET math with myth, weather won inside buy-right or left[node]outside Homer’s inner-circles.

  223. Paul Vaughan says:

    Plato’s at Turn

    I’m not giving this an exhaustive treatment — just a “suggest dove won“.

    =
    “Using an ancient Greek mathematical method described by the philosopher Parmenides, the UCL team not only explained how the cycles for Venus and Saturn were derived but also managed to recover the cycles of all the other planets, where the evidence was missing,”
    […]
    […] the mechanism’s front-dial ring is a 354-day lunar calendar, opposed to a 365-day calendar as previously suggested.
    […]
    “Our finding that the Antikythera mechanism did not have a 365-day calendar as previously thought, presents an important and exciting challenge to the research community. We have presented clear data regarding a fundamental attribute of the machine that is verifiably different to what was long held to be true, and this of course leads to a cascade of interesting and exciting consequences […]
    =

    354 = 2*(47+59+71) ——— top monster factors
    Recall that lunar cycles 18.6, 31, 62, 93 fit in 744 (Ramanujan, Heegner).

    oldmanK linked from Suggestions-44 to an article on the Antikythera Mechanism (AM).
    The authors didn’t note some obvious connections to their gearing summaries — a few examples:

    Venus
    1.59868955949705 = beat(1.00001743371442,0.615197263396975)
    1.59861315657607 = 1/beat(744/20,0.615197263396975)
    1.59861591695502 = 462 / 289
    Elaboration on the last one:
    462 = 496-28-6 = s(496)-s(28)-s(6) —– all perfect numbers
    289 = 323-28-6 = (196883-196560)-s(28)-s(6) —- monster, Leech, & perfect nos.
    1.59861591695502 = (496-28-6) / (323-28-6)

    Jupiter
    1.0920796543202 = beat(11.8626151546089,1.00001743371442)
    1.0920245398773 = 178 / 163 —- sum of baby monster factors over top Heegner no.
    1.09206349206349 = 344 / 315
    344 = 378-28-6 = sum of monster factors minus the 2 smallest perfect nos.
    315 = sum of Heegner numbers minus 1

    Jovian & lunisolar periods can be neatly tied to this scale:
    216 = 3^3+4^3+5^3 = 6^3
    1728 = 6^3+8^3+10^3 = 12^3
    884736 = 48^3+64^3+80^3 = 96^3 ——- JS phi nos. anomalistic-draconic bridge

    Recall from hexagonal number (1, 19, 37, & 61) notes above:
    427 = (196883-196560) + d(2,1/2,s(196883-196560)) = 323 + 104
    360 = (196883-196560) + s(196883-196560)
    σ(427) = 496
    σ(323) = 360 = Φ(427)
    28=(19+37)/2

    Now, pay careful attention AM fans.
    Plato’s at turn:

    3456 = (496-442)*(442-378) ——- 496 is perfect; 378 is monster sum = supersingular prime sum

    Trump apparently had lousy “climate advisors”. I suspect trolllionair mi$$guidance directing agents$ provacateurs$ to mmmisunderstand, mmmisrepresent, and mmmisrepresent with “commmandand con troll opportunismmm”. We learned as much about (bad) human nature as numbers last year.

    Trump should dove tale peacefully to dominate the middle ground with an urban feminine touch — maybe 100% cure homelessness to demonstrate that strong will can shed “the bad trolllionair mi$$guidance” to do good.

    Poor backs are breaking. The rich will have to carry “the climate burden” because they are the ones able to make the sacrifices needed in a fight they champion. Joe Trump and Don Biden work together on “70^2 dove tale”:

    Did the AM authors underscore the Plato’s number links? (I only skimmed the paper in a few minutes. Someone else can check…)

    Saturn
    1.0351711566943 = beat(29.4474984673838,1.00001743371442)
    1.03512880562061 = 442 / 427

    IT’s a maze UN what “climate advisors” curryUSsly PRmoat for poll IT talk Cal. sci11UNs.

  224. Paul Vaughan says:

    ‘Hummmor’well?well?well$0$SIM$UN$says”DO!..ugh!!”

    3600 = beat(1800,1200)
    720 = axial(1800,1200)
    1200 = beat(600,400)
    240 = axial(600,400)

    73500 = beat(1500,1470)
    49000 = beat(1000,980)
    1500 = beat(73500,1470)
    1000 = beat(49000,980)
    1200 = axial(6000,1500)
    2400 = harmean(6000,1500)
    1500 = beat(6000,1200)
    1000 = axial(6000,1200)

    6000 Herd Dull$

    Let US pray: May peace naturally ‘B with’ EURepeatedly for ‘weather UNnatural’ has heard.

  225. Paul Vaughan says:

    A crude check on implicit conventional assumptions verifies “adjustments” instinctively predicted from 04 pattern before even seeing 13 & 20 ‘up’dates:

    Totally remarkable is the sci11UNs weave herd AB out this. Using generalized wavelets that can flex to mimic the methods of others (see note above) there was no problem figuring out what assumptions others (people we trusted…too naively) made implicitly.

    Review with augmented hindsight:

    Underscores modularity. Rich layers of geometry, circulatory topology, nonlinear aliasing. No point masses (plates, layers, shells, material boundaries, flow guides, differential pulses, etc.) Countless years of work to be done by a lot of careful, patient people (ethically well-paid and free from political harassment). Maybe classy folks all-ready have ‘the trade’ secrets….rightly left a mist eerie – vanishing from “the conscious thought with adjustment”.

  226. oldmanK says:

    PV: your last posted chart -the Bond event- looked familiar. I have superimposed that chart on other data with same chronological datum. See link below.
    https://melitamegalithic.wordpress.com/2018/06/29/searching-evidence-3/

    The chart -in blue matches/is same as curve ‘B’ in the D’Andrea paper. In latter paper curve ‘C’ shows spikes in between, linking to other events. The dates 5200bce and 3200bce stand out as major events of tectonic nature. It is not just a ‘climate’ affair. What is also clear is the predominance of the 980yr cycle.

  227. Paul Vaughan says:

    Heegner Honeymoonsh…

    …shy yen nome aura11a$karing:
    18.6129709123853 = 1/LNC
    8.84735293159855 = 1/LAC
    6.40939079526111 = slip(0.999978614647502,0.0745030006844627/2)
    1.18483367770519 = slip(0.499989307323751,0.0745030006844627/2)
    2.36966735541038 = slip(0.999978614647502,0.0745030006844627)

    D-UK ace sun luke? sea$imply phi e^duke-case UN?

    19 MET tune IC = 163 mod 24 = 67 mod 24 = 43 mod 24 = 19 mod 24
    76 Cal. lip IC = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24

    Bold nos. pea & she11 game.
    37 = s(196883-196560)
    360 = (196883-196560) + s(196883-196560) = 323 + 37
    58 = 216-58-37-22-18-13-10 ———– Plato nos. Heegner: 1+2+3+7+11 = 24

    179 = 10+13+18+22+58+58 = 216-37 = 323-144 = 360-144-37 = 2*71+37 —- low NY $0 11’air

    Note$0[]we11aware phi11oss$offer next ‘tip here’.
    95 = 37 + 58
    s(95) = 25; s(25) = 6; s(6) = 6 —————————————— perfect

    The point’s perfectly balanced partition of ‘one 19’.
    3337 = 41*71; s(3337) = 119; s(119) = 25; s(25) = 6 = s(6)

    no. just what you mean” — Queens of the Stone Age “mmm[e^11]CO[2]LA” hivein11unDunZooS

    119 = 3456-41*71 = s(41*71) = 95+24
    25 = 59*59-3456 = s(3456-41*71); 144 = 3456-41*71 + s(3456-41*71)
    whooooosh!………

    ‘Cause 6000 deep states’ $sovexhaustUN, “free” list in-clued “won” DCoy:
    119 = 178-59 = 12^2-5^2 = 19+100 = 43+76 = 67+104/2 = 163-59*59+47*71+100
    ban[k]corruptshh!yen no. how the cue test list UN too average (43,67) :
    119 = 64+55 = 8^2+1^2+2^2+3^2+4^2+5^2 = 8^2+28^2-27^2

    63 = 22+18+13+10 = 163-100
    MightBe.[$]sumfearsov.DO$who-thinkIT’sno[w]thing? If $0, let US pray, Be. ‘Cause’:
    28 = average(-63,119) = s(28) —————————————————– is perfect
    316 = sum of Heegner nos.; 100 = sum of Heegner nos. each mod 24

    24 = √σ(σ(σ(σ(σ(σ(59*(163-104)-3456)))))) — “point & sh![IT]” — Queens of the Stone age “Mexicola”

    11^2 = 63+58 = 216-95 = 179-58 = (119+178-47-71)-58
    121 = average(37,323)-59 = average(-37,323)-59+37
    “setting sun deals hands of gold….” — Queens of the Stone Age
    σ(63) = 104; σ(58) = 90 = (323+37)/4
    194 = σ(63)+σ(58) = 104+90 ———————— “Sporadic and Exceptional” 194
    490 = σ(σ(194))-194 = 5/3*σ(194) = 496-6 = s(496)-s(6) —- ( $skips$write 2 the endUN )
    73 = average(19,43,67,163) = 194-11^2 ——— the low west prime can grew when 2 “won” mod 24

    Owls $sea even better in moonlight:

    640 = 744 – 104 ———————- Ramanujan:
    18.6 = 744/10/4 ———————- sol. air-seas$time
    1860 = σ(σ(216)) = σ(25*24)
    884736 = 216*64^2 = (163 – 67)^3

    1728 = 27 * 64 ————– note: 27 = 2+3+4+6+12
    24 = 64^2 – ⌊(e^√7π)⌉ = d(k,1/2,7) for k=2,3,4,6,12
    7 = 323-316 = 196883-196560-163-67-43-19-11-7-3-2-1 = 216-209

    24 is the only integer bigger than 1 with this property

    31 = 1^2+2^2+3^2+4^2+5^2 – 24 = 28^2-27^2 – 24 = average(43,67) – 24
    31 = 194-163 = σ(59*(163-104)-3456) = average(19,43) = 496/16 = 744/24
    rightly left tale IT tall misstory freeUNwe11$UN

    1.18518518518519 = beat(6.4, 1 )
    0.0745052386495925 = harmean(1.18518518518519,4/104)
    0.0748048814206895 = beat(18.6,0.0745052386495925)
    0.0808530870067242 = beat(1,0.0748048814206895)
    0.0754427537633223 = beat(8.84736,0.0748048814206895)

    True: pack-Cal. tight UN with 0.999978614647502 = 365.242189 / 365.25.

    1.18515514607101 = beat(6.4,0.999978614647502) ——– alt. route: 237 = 178+59
    27.2124530689878 = 365.25 * 0.0745036360547235
    27.3218928734485 = 365.25 * 0.0748032659095099
    29.5309517513699 = 365.25 * 0.0808513394972482
    27.5548656406489 = 365.25 * 0.0754411105835699

    comm. par[]$0[]well:
    0.0745030006844627 = 27.212221 / 365.25 ——————– draconic
    0.0748024157783867 = 27.3215823630557 / 365.25 ——— tropical
    0.0808503463381246 = 29.530589 / 365.25 ——————– synodic
    0.0754402464065708 = 27.55455 / 365.25 ———————- anomalisitc

    Joe Trump & Don Biden = 70^2 / 10 D-owning “if USA $0”, who knew clear est image UN IT?
    490 = “The Party” IT shh!UN [not $0] f(UN) shh!UN $0 v.19

    There ‘might B-19 west’ turn mess ages herd around town Be. ‘Cause sov.’ infearmazeUNwarmmmops$0[]well.

    typo: “mmmisunderstand, mmmisrepresent, and mmmisrepresent” —- WHO’s DOwing IT?
    mmm = mmmisunderstand, mmmisinterpret, and mmmisrepresent —– No. why D-y’a[11]

    “B[11]yen US D-air www(eight^2) he$ave UN — 11ed22plan B-11ink UN PR Joe Act-Rome: UN no morals

  228. Paul Vaughan says:

    7200 Year Lunisolar

    “…and a11$he$aid was true:” — QOTSA “Mexicola”

    x	y	beat	axial	harmean
    400.2971813	600.2413963	1201.709883	240.145602	480.291204
    1200.891544	1800.724189	3605.129648	720.436806	1440.873612
    400.2971813	1200.891544	600.445772	300.222886	600.445772
    400.2971813	1800.724189	514.7178777	327.4956013	654.9912027
    600.2413963	1200.891544	1200.074319	400.2063374	800.4126748
    600.2413963	1800.724189	900.3620944	450.1810472	900.3620944
    

    7200 / preceding:

    x	y	beat	axial	harmean
    17.98663677	11.99517402	5.991462752	29.98181078	14.99090539
    5.995545589	3.998391339	1.997154251	9.993936928	4.996968464
    17.98663677	5.995545589	11.99109118	23.98218236	11.99109118
    17.98663677	3.998391339	13.98824543	21.98502811	10.99251405
    11.99517402	5.995545589	5.999628427	17.99071961	8.995359803
    11.99517402	3.998391339	7.996782677	15.99356535	7.996782677
    

    “…a ‘one door’ that those [funds] don’t point at [clue]” — Queens of thus-tune neige

  229. Paul Vaughan says:

    oldmanK ( March 23, 2021 at 10:51 am ) emphasized 980.

    There’s quite a lot more than just 980 to note.

  230. Paul Vaughan says:

    Hear Comms. The…

    OB V-yes can’tr[us]t see1eve1800.

    6000 / 1stable AB[d]ove:

    x y beat axial harmean
    14.98886397 9.995978347 4.992885627 24.98484232 12.49242116
    4.996287991 3.331992782 1.664295209 8.328280773 4.164140387
    14.98886397 4.996287991 9.992575982 19.98515196 9.992575982
    14.98886397 3.331992782 11.65687119 18.32085676 9.160428378
    9.995978347 4.996287991 4.999690356 14.99226634 7.496133169
    9.995978347 3.331992782 6.663985564 13.32797113 6.663985564

    “DO’$what’ ever ‘US pie’ D-air can” — Spied Air Man[]

  231. Paul Vaughan says:

    The OBvious Scale$ Odd $now:

    x y beat axial harmean
    400 600 1200 240 480
    1200 1800 3600 720 1440
    400 1200 600 300 600
    400 1800 514.2857143 327.2727273 654.5454545
    600 1200 1200 400 800
    600 1800 900 450 900

    1470 / preceding:

    x y beat axial harmean
    3.675 2.45 1.225 6.125 3.0625
    1.225 0.816666667 0.408333333 2.041666667 1.020833333
    3.675 1.225 2.45 4.9 2.45
    3.675 0.816666667 2.858333333 4.491666667 2.245833333
    2.45 1.225 1.225 3.675 1.8375
    2.45 0.816666667 1.633333333 3.266666667 1.633333333

    $aid better with numbers than words: won decade ago.

  232. Paul Vaughan says:

    7200$he$in$rich

    x	y	beat	axial	harmean
    400	600	1200	240	480
    1200	1800	3600	720	1440
    400	1200	600	300	600
    400	1800	514.2857143	327.2727273	654.5454545
    600	1200	1200	400	800
    600	1800	900	450	900
    

    7200 / preceding:

    x	y	beat	axial	harmean
    18	12	6	30	15
    6	4	2	10	5
    18	6	12	24	12
    18	4	14	22	11
    12	6	6	18	9
    12	4	8	16	8
    
  233. Paul Vaughan says:

    Berg-Queens$Sov. West Turn UN IT

    24 = √σ(σ(σ(σ(σ(σ(25)))))) — “setting$UNdeal$handsovgold…”

    “in NA-world that’
    SFu11 of sh!IT:
    UN graphs
    a11ign”
    — Queens of D-US$Tuneag[//]e

    Rome: UN no morals??? ( “IT’s a ‘won’ D-air that…..” )

    24 = √σ(σ(σ(σ(σ(σ(5^2)))))) —— “and a11[11]$aidWA$true” — Queen$sov.J-US$Tone-neige “mmm[e/11]CO[2]LA”

  234. Paul Vaughan says:

    Left[node]out

    1470 = axial(73500,1500)
    980 = axial(49000,1000)

    x y beat axial harmean
    400 600 1200 240 480
    1200 1800 3600 720 1440
    400 1200 600 300 600
    400 1800 514.2857143 327.2727273 654.5454545
    600 1200 1200 400 800
    600 1800 900 450 900

    6000 / preceding:

    x y beat axial harmean
    15 10 5 25 12.5
    5 3.333333333 1.666666667 8.333333333 4.166666667
    15 5 10 20 10
    15 3.333333333 11.66666667 18.33333333 9.166666667
    10 5 5 15 7.5
    10 3.333333333 6.666666667 13.33333333 6.666666667

    comparative interpretation:
    6000 without 1800
    7200 with 1800

    Spook EU’11ace $UN:
    1. NATO & Russia together$hood’VE D-taled mode11ove DO, Bond, & Heinrich EVents.
    2. Neither seas solar cycle length aggregation CR IT eerie awe $0 Φicy UN t(ally) …well.

    Lunisolar
    16.8627856518082 = beat(18.6129709123853,8.84735293159855)
    491.132481334807 = slip(179.333323110834,16.8627856518082)

    JEV
    491.049532939153 = slip(146.000401793899,44.2784629967674)

    146.000401793899 = slip(44.2784629967674,1.59868955949705)
    73.0002008969497 = slip(22.1392314983837,0.799344779748523)

    5256 = 7920-2400-240-24 ——– M11, E8
    5256.63939957554 = slip(44.2784629967674,0.0952207761715205)

    Puttin’ dem on defense?…

    360 = 37+323 ———————– Monster, Leech
    360.180290240748 = slip(146.000401793899,22.1392314983837)

    …$sieve tooo gather.

    180 = AVERAGE(37,323)
    180.090145120374 = slip(73.0002008969497,11.0696157491919)

    Lo:LAs[n]iff there’s any miss story left ABout D-meaning of “monster US moon [sh!]ine”.
    Can JSUN play a winning hand? No. doubt….(“DON’T$PEAK”….) $0 11ong JSUN! EU failed.

  235. Paul Vaughan says:

    Bon Nos. $Sum Ware in the “Original 39”, USA?

    Image: UN IT in spy Eur. ring “Ramanujan’s Encryption” of sol air-seas time order.

    39 = MOD(323,71)
    39 = 331-292 = 331-163-67-43-19
    39 = 354-315 = 354-316+178-47-59-71 = 47+59+71-316+178

    178 = baby monster sum
    316 = Heegner nos. sum
    177 = top monster (supersingular) sum = 47+59+71
    292 = top Heegner sum = 19+43+67+163

    322.996026127048 = 100*ΦΦ/(1/11.8626151546089+1/29.4474984673838)
    26253410.1345891 = 1/(φφ*(1/11.8626151546089+1/29.4474984673838)/100-1/323)
    10 = ⌊LOG_10 (262537412640768744 / 26253410.1345891)⌉
    26253741 = ⌊262537412640768744 / 10^10⌉
    331 = 26253741 – 26253410
    331 = 163+67+43+19+MOD(196883-196560,71) —— this is “The Remain D-air”
    ⌊262537412640768744 / 10^10 – (163+67+43+19 + (196883-196560) mod 71) ⌉

    322.996026127027 = axial(26253410,323)
    8.45614574631707 = 322.996026127027*φφ/100
    8.4561457463176 = axial(29.4474984673838,11.8626151546089)
    -0.000000000006 = % error

    39 = 64 – 25 = 8^2 – 5^2
    39 = (331-323)^2 – (331-2*163)^2
    331 = 8 + 323
    331 = 5 + 2*163

    φ(111) = 8^2 – 5^2 = 39
    171.406220509606= 111.5/(5.5*(J+S)+1/11500)
    171.406220601552 = beat(164.791315640078,84.016845922161)
    -0.000000053642 = % error

  236. Paul Vaughan says:

    The √2 Deep Hole

    “dr. 11!” led US astray…

    “[…] the covering radius of the Leech lattice is √2; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least √2 from all lattice points are called the deep holes of the Leech lattice. There are 23 orbits of them […]”

    Review: Lunisolar deep hole was drilled with √2, φ & 96^3 on Suggestions-43. “First order” simplifications appear above. Weave time to explicitly cover ~0% of the SIM MET trees dial UN in the (no longer mysteriously-named) monstrous moonshine.

  237. Paul Vaughan says:

    “Sea Buy”$peakTrumPReview: Vasiliev Dergachev 2002 14C Bispectrum

    Recall amiss story US “curious” IT‘$noted earlier.

    V&D suggested 2400 splits 710 into 570 & 940
    547.909967845659 = axial(2400,710)
    1008.28402366864 = beat(2400,710)
    …whereas 2900’s the number that does that.

    570.360110803324 = axial(2900,710)
    940.182648401826 = beat(2900,710)

    We looked at (19 mod 24) JS slip cycle (can D)-dates (in pari$11ol). One‘$now restated 4 comm. Paris$UN with 11’UN-US$0lure $he$aveUN (note well what’$PRint’don back Sov. $19 bill) :

    574.604095118134 = axial(2959.13381403679,713.067400275117)
    ThePRsentacesun$notPRoofUS$UNa11″enough!”tooSIMbilllies “the baby monster” (wtf did you think was going on?…lol)
    939.447668560927 = beat(2959.13381403679,713.067400275117) — compare what follow$

  238. Paul Vaughan says:

    “Le Ruse Sea 940” B-est 2 [Mute]

    Y(e/11) “UN$careSFull!”
    too seas aqua [11]INO:
    what $ain’t N(IC)
    Auriga
    “$no. don!e”

    Pole IT test isn’t $0 SIM pully “ad just in can’t afford DO tool $saveUNgrrace:

    5.99685290323073 = beat(0.0754402464065708,0.0745030006844627)
    diversity:
    1986.46983643213 = slip(5.99685290323073,0.999978614647502)
    1844.25352645575 = slip(5.99685290323073,1.00001743371442)
    1814.31362251033 = slip(5.99685290323073,1.00002638193018)
    unity:
    1878.75398044817 = harmean; 939.376990224084 = harmean / 2
    1878.75398044786 = slip(5.99685290323073,1.00000747633418)

    Hole “D-UN there” cards ever $0 Close 2 Ratchet most in$π-red D-est$ware “frontline con” text.

    2.99842645161536 = beat(0.0377201232032854,0.0372515003422313)
    diversity:
    993.234918216065 = slip(2.99842645161536,0.499989307323751)
    922.126763227873 = slip(2.99842645161536,0.500008716857211)
    907.156811255165 = slip(2.99842645161536,0.500013190965092)
    unity:
    939.376990224084 = harmean
    939.376990223931 = slip(2.99842645161536,0.500003738167089) —– ($alt.turn net$CR(US|e) )

    “What ever!est π-DO Or[We11]Can ” — spied air-mmman[EU ] will

    1879.26478947996 = slip(179.333323110834,5.99685290323073)
    1879.26478947996 / 2 = 939.63239473998

    PRscript$UN “D-note’$in turn a11′ imageUNair rise“:

    0.0374012078891933 = axial(0.499989307323751,0.0404251731690623)
    9.30648545619264 = beat(0.0374012078891933,0.0372515003422313)
    4.42367646579927 = beat(0.0377201232032854,0.0374012078891933)
    2.99842645161536 = axial(9.30648545619264,4.42367646579927)
    89.6666615554168 = slip(9.30648545619264,4.42367646579927)
    939.63239473998 = slip(89.6666615554168,2.99842645161536) ——– (Homer no$. )

    B[11]art 11’UN-US $0 lure “$19B[11]lies” con-f(US)yen mag (net IC) i.e.: luke sea
    tames comm. red$ain’t no. baby monster play pen (Merge: 44sea28 Be. low ).

    NearAD[11]ove tale$now EU-wood B-leave[?] IT’s D-liberate con f(use)UNframmeUN[mic:CRowe$]offline[sci11UNs].

    WHO’s $19 UN bill “luck CA11 Cairo” PR actors f(air) 43(sea)70 what’s next “in ever give UN $sue west turn$ can AI”? D-ON lie pla[y]TO no. $can[D]a11ove green NAtoo:

  239. Paul Vaughan says:

    Which Typo

    “Suggestion: Study Schneider’s delightful proof and strive to understand why replacing φ (which gives e) with 240 [typo] gives Earth’s sidereal orbital period so precisely.”

    73 = average(19,43,67,163) = average(104,42) = 194-11^2
    11^2 = 63+58
    194 = σ(63)+σ(58)
    63 = 22+18+13+10 = 163-100; 100 = sum of Heegner nos. each mod 24 = 1+2+3+7+11+19+19+19+19
    58 = 216-58-37-22-18-13-10

    490 = σ(σ(194))-194 = 5/3*σ(194) = 496-6 = s(496)-s(6) = partition function of 19
    4900 = 70^2 = 24^2+23^2+22^2+…+3^2+2^2+1^2

    95 = 37+58
    s(95) = 25; s(25) = 6
    σ(95) = 120; σ(120) = 360 = (196883-196560) + s(196883-196560) = 323 + 37

    265 = 360 – 95 = 323 – 58; s(265) = 59
    s(120) = 240 = 216 + √σ(σ(σ(s(47+59+71)))) = 216 + √σ(σ(σ(63)))
    s(240) = 504 = 744 – 240; σ(240) = 744 = 504 + 240

    “That’s 0.597351554461056 seconds per century.”

    σ(63) = 104; σ(104) = 210 = 7# = 7*5*3*2*1
    σ(210) = 576 = 24^2 = 4*144 = 4*(59*59-47*71)

  240. Paul Vaughan says:

    MonsterUSAuthorIT

    Review

    4428 = 4370 + 58 = 4370+216-58-37-22-18-13-10

    37 = s(196883-196560)
    323 = (196883-196560)
    360 = (196883-196560)+s(196883-196560) = 323 + 37 = σ(196883196560) = σ(323)

    104 = d(2,1/2,37) = d(4,1/2,58) = d(2,1/2,58)
    4428 = 8128 – 3700; 8128 = s(8128) —- perfect no.
    104 = average(-7920,8128); 7920 = M11 order

    Crude

    29.4541077985279 = ΦΦbeat(104,44.28)
    11.8626876026982 = ΦΦaxial(104,44.28)

    323 ~= 100ΦΦΦΦ/(1/beat(104,44.28)+1/axial(104,44.28))

    22.14 = 1/(1/beat(104,44.28)+1/axial(104,44.28)) = 44.28 / 2
    52 = 1/(1/beat(104,44.28)-1/axial(104,44.28)) = 104 / 2

    Refined

    298 = σ(323)-104+42 = 360-104+42 = 19+43+67+163+216-58-37-22-18-13-10 – 52
    240 = 496-256 = σ(323+104)-σ(323)+104 = 19+43+67+163 – 52

    58 = 298-240 = 216-58-37-22-18-13-10

    58 = 100 – 42 = sum of Heegner nos. each mod 24 – 42 = 19+19+19+19+11+7+3+2+1 – 42
    58 = 104-((average(59*59,-47*71))^5-67^5-43^5-19^5-47^5)^(1/5)
    58 = 104-average(-71,163)

    Lunisolar “Spider Web$” N0! D-ou[B]t

    490.178476738918 = √(400.297181344069*600.241396282931) —- geometric mean
    22.1399746327524 = √490.178476738918
    4427.99492655047 = 200 * 22.1399746327524
    4428 = ⌊4427.99492655047⌉

    1470.53543021674 = √(1200.8915440322*1800.72418884878) ——— geomean
    490.178476738915 = 1470.53543021674 / 3
    22.1399746327523 = √490.178476738915
    4427.99492655046 = 200 * 22.1399746327523
    4428 = ⌊4427.99492655046⌉

    B: 4370 = 4428 – 58

  241. Paul Vaughan says:

    Play DO$19 bill * 4 UN IT

    164.791315640078 +
    84.016845922161 +
    29.4474984673838 +
    11.8626151546089 +
    1.88084761346252 =
    291.999122797694 ——————– no.%-error

    – 1.88084761346252 +
    1.00001743371442 +
    0.615197263396975 +
    0.240846697327135 =
    291.97433657867 ———————- $0 11air $seas$ time$ alter-net coarse

    292 = 163 + 67 + 43 + 19
    292 = 163mod24 + 67mod24 + 43mod24 + 19mod24 + 216 = 19+19+19+19+216 = 76+216

  242. Paul Vaughan says:

    Jupiter-Saturn Offset$

    600 = σ(216)
    1800 = σ(1691) = σ(760)
    400 = s(2401) = s(496+28) = s(836-42); σ(σ(400)) = 993 = 1986/2
    1200 = σ(378+178-100) = 744+378+178-100
    744 = s(378+178-100) = 1200-378-178+100

    perfect recall: 28 = s(28); 836 = lowest weird untouchable no.
    378 = sum of supersingular primes = monster factors sum
    178 = baby monster (B) factors sum
    100 = sum of each Heegner no. mod 24

    Monster “US$ Heegner” — Some Loop Hale$

    73 = s(378-43) = s(s(993)) = s(s(1986/2)) = average(19,43,67,163) = average(42,104)
    49 = s(378-163) = s(378-19-59*59+47*71) = 7^2 = 73-11-7-3-2-1 = 73 – 24
    95 = s(378+67) = 37+58 = 216-58-22-18-13-10
    multiperfect recall s(95) = 25; s(25) = 6 = s(6); σ(95) = 120; σ(120) = 360
    360 = σ(378-19) = σ(378-163+59*59-47*71)
    7 = √s(378-163) = √s(378-19-59*59+47*71) = 323-316 = 196883-196560-(Heegner sum)

    Wood $UN B-ring perfect lunisolar record$ :

    400 = σ(496-s(378-163)-104)
    600 = σ(496-s(378-163))
    1200 = σ(2*(496-s(378-163)-104))
    1800 = σ(2*(496-s(378-163)))

    Aim$Bond DOn’t $peak: average sea cure IT count$sol node 13*7.2=93.6out “Sporadic & Exceptional” McKay & He in rich.

  243. Paul Vaughan says:

    Semi-Perfect “Bei. B No$.” 940

    σ(1879) = 1880
    σ(1880) = 4320 —- Plato $wing from the Chand11ier — C!A

    216 = σ(average(-178,496))
    4320 = average(4370,4270) = average(4370,s(4370))

    2 is 3-hemiperfect; 24 is 5-hemiperfect
    4320 = lowest 7-hemiperfect no.

    7 = 323-316 = 196883-196560-163-67-43-19-11-7-3-2-1
    “There’s more to the picture than meets the [sky]” — Neil Young

  244. Paul Vaughan says:

    Ally Caught Some Chande/11ier$wingUN

    DCice$sovCO[2]UN[T]sol 13 * 7200 with average in form ace yen wharf…..

    4370 = 4370
    s(4370) = 4270
    s(4270) = 4658
    s(4658) = 2794
    s(2794) = 1814
    s(1814) = 910
    s(910) = 1106
    s(1106) = 814
    s(814) = 554
    s(554) = 280
    s(280) = 440
    s(440) = 640
    s(640) = 890
    s(890) = 730
    s(730) = 602
    s(602) = 454
    s(454) = 230
    s(230) = 202
    s(202) = 104
    s(104) = 106
    s(106) = 56
    s(56) = 64
    s(64) = 63
    s(63) = 41
    s(41) = 1
    s(1) = 0

    sum = 26193

    433 = 26193 – 25760
    432 = average(4370,s(4370)) / √(4370-s(4370))

    25760 = ⌊25760.434943341⌉ = ⌊26 * beat(1.18483367770519,1.18341848397804)⌉
    25760 = ⌊25760.4349434063⌉ = ⌊beat(1.00001743371442,0.999978614647502)⌉

    Tropical (semi-annual & nodes (semi-draconic))
    1.18483367770519 = slip(0.499989307323751,0.0372515003422313)
    433 = ⌊432.76050078182⌉ = ⌊365.25 * 1.18483367770519⌉

    Sidereal
    1.18341848397804 = slip(0.500008716857211,0.0372515003422313)
    432 = ⌊432.243601272978⌉ = ⌊365.25 * 1.18341848397804⌉

    “[D-air] $’more [’22] the pick t[ru]e than meets the [sky]…” — Neil Young

    QBO equivalent:
    865.521001563641 = 365.25 * 2.36966735541038
    866 = 2*(26193-25760)
    864.487202545956 = 365.25 * 2.36683696795607
    864 = (4370+s(4370)) / √(4370-s(4370))

    ….. “hey, hey, My My …” — NY …..

    …..air is sol T.

  245. tallbloke says:

    “hey, hey, My My …”
    There’s more to the picture,
    Than meets the eye.

    Not a fan of Neil Young’s politics myself, but I can actually play this song on my guitar!

  246. Paul Vaughan says:

    Introduction: Tip of Lunisolar Confounding Iceberg

    6.07098113178251 = beat(491.132481334807,5.99685290323073)
    72.9984824166731 = slip(19.8650360864628,6.07098113178251)
    208.899107282611 = slip(72.9984824166731,9.93251804323141)
    208.886643858908 = JS slip cycle

    Nominal comparisons:

    6.07438016528926 = beat(490,6)
    73.5 ~= 73.4929686622967 = slip(19.8650360864628,6.07438016528926)
    210 = slip(73.5,10)

    6.07420653406045 = beat(491.132481334807,6)
    73 ~= 73.5 ~= 73.4675602385511 = slip(19.8650360864628,6.07420653406045)

    6.07115457866136 = beat(490,5.99685290323073)
    73 ~= 73.0235673639375 = slip(19.8650360864628,6.07115457866136)
    209 ~= 208.812149943122 = slip(73,9.93251804323141)

    Review comparatively:

    6.01674004100847 = beat(1814.31362251033,5.99685290323073)
    65.8594372352303 = slip(19.8650360864628,6.01674004100847)
    50.0701067244605 = slip(19.8650360864628,3.00837002050424)
    96.1935351354127 = slip(19.8650360864628,1.50418501025212)
    208.849200346502 = slip(65.8594372352303,19.8650360864628)
    178.330515859016 = slip(65.8594372352303,9.93251804323141)
    503.940520714068 = beat(356.661031718033,208.849200346502)

    0 slip at:
    491.122639395668
    1815.25586361718

  247. Paul Vaughan says:

    744’s “Only Won” Seas$[360]UN

    738 = -d(3,1/2,11) —– 39 no. mystery “well B low”
    745 = d(3,1/2,16) —— too dem-miss[TX]tify
    —————————————–
    744 = d(3,1/2,28) —- perfect no. (s^1)(28) tie one-and-only-won anti-aliquot (s^(-1))(28)
    —————————————–
    744 = -d(3,1/2, 19)
    744 = -d(3,1/2,43)
    744 = -d(3,1/2,67)
    744 = -d(3,1/2,163)

    Φ-command air-laps past.

    Biden, “Don seas trees”. “UN Trump”, Joe putters $at turn quest yen:
    19.8650360864628 = beat(29.4474984673838,11.8626151546089) — sidereal
    19.8588720868409 = beat(29.42351935,11.85652502) ————- tropical
    64000.2003306175 = beat(19.8650360864628,19.8588720868409)

    Overseas navy: We blue secure IT.

    640=744-104
    320 = average(744,-104)
    320=19+43+67+163+28
    360 = σ(19+43+67+163+28-378+178)

    “can keep EU-WA11 2 mmm-ice[sh]elf; DO EU wanna B free? ‘$0 free’ Eur. S[H]elf” — Queen: $solve THUS tune-neige

  248. Paul Vaughan says:

    Earth’s Most Prime AI-insight$240

    1.00001743371442 = 1/E
    1.00001743390371 = (1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)
    1.00001745798498 = average(-178,378)*2.92005097731613/(19+43+67+163)

    Seed the algorithm as follows to reach all primes up to exactly 19 and no further:
    2.92005090699884 = (19+43+67+163)/average(-178,378)*(1-(1/240)^1)^(0/1)/(1-(1/240)^2)^(2/2)/(1-(1/240)^3)^(3/3)/(1-(1/240)^4)^(2/4)/(1-(1/240)^5)^(5/5)/(1-(1/240)^6)^(1/6)
    2.92005090644612 = 1.00001743371442*(19+43+67+163)/average(-178,378)

    Repeating as many times as necessary:
    “Suggestion: Study Schneider’s delightful proof and strive to understand why replacing φ (which gives e) with 240 [typo] gives Earth’s sidereal orbital period so precisely.”
    “That’s 0.597351554461056 seconds per century.”

  249. Paul Vaughan says:

    Janko: “Monster tie” when Lyons “the original 39” Leech

    70 = √(24^2+23^2+22^2+…+3^2+2^2+1^2)
    71 = Δ(111)
    72 = average(71,73) = average(59*59,-47*71) = Φ(111) = Φ(73)
    73 = average(42,104) = average(19,43,67,163) = φ(1333)
    5256 = 72*73 = 7920-2400-240-24

    144 = 2*72 = 2*Φ(111) = 288/2 = Φ(1260)/2
    144 = 12^2 = 59*59 – 47*71 = 71 + 73
    144 = average(-(71^2),73^2) = average(-(Δ(111)^2),φ(1333)^2)
    144*4 = 576 = 24^2 = 70^2 – (23^2+22^2+21^2-…+3^2+2^1+1^1)

    288 = 73^2-71^2 = φ(1333)^2-Δ(111)^2 = Φ(323) = 2*Φ(111) = Φ(Φ(1333))

    φ(1333) = 73
    Φ(1333) = 1260
    Φ(1260) = 288 = 4*72 = 4*Φ(111) = Φ(Φ(1333))
    Φ(111) = 72
    φ(111) = 39

  250. Paul Vaughan says:

    111 “snow done” miss story lie UN$he ignore

    -378 = -2-3-5-7-11-13-17-19-23-29-31-41-47-59-71
    59 = average(-378,496)

    111 = 378+Φ(196883-196560)-496-59
    111 = Φ(323)-(47+59+71) = Φ(196883-196560)-(47+59+71)

    163 = 2+3+5+7+11+31+37+67 = Ly factor sum of Lyons group
    316 = Heegner number sum

    111 = 163 – 104/2 = 316-67-43-19-11-7-3-2-1-104/2
    111 = 153 – 42 = 316-163-42

    Δ(111) = 71 = top monster factor[well]

  251. Paul Vaughan says:

    Care Actor$[194]Table

    “The character table of the monster, a 194-by-194 array […]”

    Review: 360 = 323+37. Study the squares around the top of the monster.
    323 = 196883-196560
    37 = s(323) = average(-104,178); 104 = -d(2,1/2,37)

    71^2 = 73^2 – Φ(323)
    71^2 = 70^2 + 141 —————————– 141 ± 37
    71^2 = 70^2 + 378-178-59

    141 = average(104,178) = average(42,240) = 3*47 = 104+37 = 178-37
    141 = average(19+43+67,1+2+3+7+11+19+43+67) = average(129,153)
    141 = 378-178-59 = 200-59

    292 = 104+178+163-153; average(104,178,163,-153) = 73

    McKay & He left 104 & 42 a mystery. “can go with the flow: DO EU B-leave IT in Eur. head? IT’s$not safe 2 play along — QOTSA

    May peace be with you.

  252. Paul Vaughan says:

    6000 Sense 1800 Janko

    X: 8, 12, 20, 44, 68, 164 = 4*B —————- caution: mixing symbols here — sea B low
    Y: 8, 12, 20, 20, 20, 20 = X mod 24 —— “setting $UN deals[]and$sov. goaled” — qotsa
    Z: 19, 19, 19, 19 = I mod 24
    sums
    X: x = 316
    Y: y = 100 = average(378,-178)
    Z: z = 76

    Some “won” nos. EU[]R lucky 2 remember: 240 ± 76; σ(240) = 744

    σ(745) = 900

    R(3,1/2,16) = 744.68686334684 = ⌊(e^√16π)^(1/3)⌉^3 – e^√16π
    745 = d(3,1/2,16) = R(3,1/2,16) – R(1,1/2,16)

    Φ(1333) = 1260; φ(1333) = 73 = average(19,43,67,163)
    1333 = Φ(1333) + φ(1333) = 1260 + 73

    “no. just what EU mean” — qotsa

    R(3,1/2,11) = -738.143065592427 = ⌊(e^√11π)^(1/3)⌉^3 – e^√11π
    738 = -d(3,1/2,11) = R(1,1/2,11) – R(3,1/2,11)

    4368 = 42*104 = σ(σ(738)) = σ(1638) = σ(1260) = σ(Φ(1333)) = σ(1500)
    1638 = 42*39 = σ(738)

  253. Paul Vaughan says:

    The Original 39 Peace

    keeper $cantrust:
    240 = 61^2 – 59^2 ————————— (key to differentiating “982” from 980)
    240 = mod(1728,744) ———————- “and all [11]’$aid WA$[SH!]true” — qotsa

    61 = 100 – 39
    100 = average(378,-178)

    378 = 2+3+5+7+11+13+17+19+23+29+31+41+47+59+71
    178 = 2+3+5+7+11+13+17+19+23+31+47

    200 = 378 – 178
    Honeymoonshine: Baby knows some luck[]key.
    194 = 378 – 178 – 6 = 200 – d

    A: 1, 2, 3, 7, 11, 19, 43, 67, 163 = Heegner nos.
    B: 2, 3, 5, 11, 17, 41 = Euler’s “lucky” nos.
    C: 7, 11, 19, 43, 67, 163 = 4*B-1
    D: 1, 2, 3 = A not in C
    E: 1, 2, 3, 7, 11, 19, 19, 19, 19 = A mod 24
    F: 2, 3, 5, 11, 17, 17 = B mod 24
    G: 7, 11, 19, 19, 19, 19 = C mod 24
    H: 1, 2, 3, 7, 11 = subset of A summing to 24
    I: 19, 43, 67, 163 = subset of A not in H = subset of A with mod 24 = 19 (see E)
    J: 163 = top Heegner no.
    K: 1, 2, 3, 7, 11, 19, 43, 67 = Heegner nos. non[$]top

    sums
    A: a = 316
    B: b = 79
    C: c = 310
    D: d = 6 = 1 + 2 + 3 = 3 + 2 + 1 ——— “Sporadic & Exceptional” — McKay & He (2015)
    E: e = 100
    F: f = 55
    G: g = 94
    H: h = 24
    I: i = 292
    J: j = 163
    K: k = 153

    C!A“$wing from the ChanDO[!]11ears”:
    316 = a = j+k = 163+153 = 4*b = 4*79
    158 = a/2 = 316/2 = average(j,k) = average(163,153) = 2*b = 2*79
    158 = 58+37+22+18+13+10 = 104-yielding levels sum = 100+58 = 216-58
    79 = a/4 = 316/4 = average(j,k)/2 = average(163,153)/2 = b = 79
    42 = 100-58 = 200-158 — 744 D-tales+11+16 above 320 note added symmetry of 738 & 745.

    365 = c+f = 310+55
    194 = 100+94 = e+g = 100+70+24 = 200-6
    216 = 310-94 = c-g = 316-100 = a-e = 158+58
    271 = 216+55 = c-g+f = a-e+f = c-f+h = 310 – 39

    recall:
    55 = 1^2+2^2+3^2+4^2+5^2 = 28^2-27^2 fits inside of:
    70 = √(24^2+23^2+22^2+…+5^2+4^2+3^2+2^2+1^2)
    194 = 24 + 70 + 100
    24 is the only integer bigger than 1 with this property
    194 = 24+√(24^2+23^2+22^2+…+5^2+4^2+3^2+2^2+1^2)+4370-s(4370)
    194 = 378-178-3-2-1 where 3, 2, 1 are from set D ….left over from EU1ure$1o[]ckey$heignore[$0 B IT!]union

    39 = g-f = 94-55 = 24+70-55 —- well-balanced lunisolar SIM MET tree
    39 = 24+√(24^2+23^2+22^2+…+5^2+4^2+3^2+2^2+1^2)-(1^2+2^2+3^2+4^2+5^2)

    No matter your nation, race, and politics: may peace be with you.

  254. Paul Vaughan says:

    Crude Lunisolar Summary

    
    x	y	lcm	gcd	harmean	geomean	ave(har,ave)	average	double	double	double	double	939.63239473998	939.63239473998	939.63239473998	939.63239473998	"710"	"710"	"710"	"710"
    400	600	1200	200	480	489.897948556635	490	500	960	979.795897113271	980	1000	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D
    400.297181344069	600.241396282931	1200	200	480.291204019241	490.178476738917	490.280246416371	500.2692888135	960.582408038483	980.356953477835	980.560492832741	1000.538577627	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D	axial V&D
    1200	1800	3600	600	1440	1469.69384566991	1470	1500	2880	2939.38769133981	2940	3000	708.482130525904	712.021034743886	712.056957839865	715.522897005214	568.604903604626	573.177647999421	573.224207677047	577.72990518934
    1200.8915440322	1800.72418884878	3600	600	1440.87361205772	1470.53543021674	1470.8407392491	1500.80786644049	2881.74722411543	2941.07086043349	2941.68147849821	3001.61573288098	708.587817831591	712.119756130532	712.155548991307	715.614771729339	568.74106544966	573.305606994134	573.352005717902	577.849706882783
    1201	1801	2163001	1	1441.0399733511	1470.71445223062	1471.01998667555	1501	2882.0799467022	2941.42890446123	2942.0399733511	3002	708.607932833041	712.140745116588	712.176557791282	715.636611058323	568.766983321465	573.332814928693	573.379241008065	577.878187450253
    

    Recall:
    73500 = beat(1500,1470)

  255. Paul Vaughan says:

    Tie $USqotsawan: N/A

    Norton called it “the voice of God” because once well-placed, buckets overflow with numeric torrent sov. insight. Revealing terse classification theorem proof can$save US from pick UN ~0% of the numeric ∞ lifetimes afford for $sharing.

    “another heart has made the trade
    DOn’t understand how a heart is a spade?

    94 = 55+Δ(55) = (1^2+2^2+3^2+4^2+5^2)+Δ(1^2+2^2+3^2+4^2+5^2) = 55+39 = 24+70

    sum how: the vital connection is made” — Elastica

    24 & 5 mark the partition boundaries:
    39 = 24+√(24^2+23^2+22^2+…+5^2+4^2+3^2+2^2+1^2)-(1^2+2^2+3^2+4^2+5^2)
    39 = 64 – 25 = Δ(240)-Δ(70) = δ(δ(194)) = 58-19 = ΣΦ(70) —– E8, Leech, monster character table

    We no. WHO take$scare of public awareness$sov. IT:
    58 = 39+19 = 216-58-37-22-18-13-10
    Adds up to “better learn to know good” from “bad rich” symbols printed on ‘the back sov.’ $19 bills (D-liberate mmm = misunderstanding, misinterpretation, misrepresentation is how sum AI is programmed too give add vice to ‘IT’s lead D-airs$’).

    58±5 = (53,63)
    153 = 1+2+3+7+11+19+43+67
    163 = 163
    100 = 1+2+3+7+11+19+19+19+19
    53 = 153-100
    63 = 163-100

    23 = Δ(39) = Δ(Δ(240)-Δ(70)) = Δ(δ(δ(194))) = Δ(ΣΦ(70))
    top factor of dimension of minimum faithful representation of baby monster

    $0 Homer knew world nos. order SIM bill US$ m[$]m in can’t talk key….

  256. Paul Vaughan says:

    Cure US IT at what cost? $19 bill yen

    elastica: “another heart has made the grade
    understand how the last card is played”

    146 = 70+19+19+19+19
    152 = 70+19+19+19+19+3+2+1
    298 = 70+19+19+19+19+70+19+19+19+19+3+2+1
    194 = 24+70+19+19+19+19+24
    104 = 70+19+19+19+19+3+2+1-24-24
    24 = 1+2+3+7+11

    Timely review of what’s now ancient history:
    DOn’t USteal my$UNshineD-light

    298±58:
    298 = average(240,2*178)

    298±53:
    298 = average(378-178+1+2+3+7+11+19+43+67,ΣΦ(163))

    Top Peacekeepers no. the importance of math.

    24 = δ(δ(76)) = δ(δ(δ(δ(292))))
    1+2+3+7+11 = δ(δ(19+19+19+19)) = δ(δ(δ(δ(19+43+67+163))))

    Miss story left Schneider & He to right terse proof of classification theorem = great test achievement in human history.

  257. Paul Vaughan says:

    IT‘s Read Word Air DO We

    “The study of psychills reveal$too US R ‘rig no. rants’ and is the[T]air4 very D-US$tar ban for ‘peep hill WHO’sideasar CR US DO 11 i.e. $D.”

    IT’$0 whirled wild deep state of ‘math We no.’ $0[]well now:
    4428 = 4370 + 19 + 39 = 4370+216-58-37-22-18-13-10

    for terrestrial tropical year length = 0.999978614647502 = 365.242189 / 365.25 and
    11.8626151546089 = 1/J
    29.4474984673838 = 1/S

    1368.23312313647 = (4370-s(4370))φ/(J+S)
    2999.612648502 = beat(1368.23312313647,939.63239473998)

    2736.46624627294 = (378-178)φ/(J+S)
    5999.22529700399 = beat(2736.46624627294,1879.26478947996)

    comm. paris $UN for terrestrial tropical year length = 0.99997862 —— same comparison repeated
    3008.30696170427 = beat(1368.23312313647,940.483841064968) —— in examples below
    6016.61392340853 = beat(2736.46624627294,1880.96768212994)

    no.mmmUNa11i.e.6000 “mmmUN$tar$inD[$]pairUS$sol” – qotsa

    1368.24995673243 = (196883-196560)φφφ
    2736.49991346486 = 2*323*φφφ

    2999.5317445019 = beat(1368.24995673243,939.63239473998)
    3008.22558803381 = beat(1368.24995673243,940.483841064968)
    5999.0634890038 = beat(2736.49991346486,1879.26478947996)
    6016.45117606762 = beat(2736.49991346486,1880.96768212994)

    2736.65450218453 = 4428Φ
    1368.32725109227 = 4428Φ/2

    2999.16034139499 = (1368.32725109227)*(939.63239473998) / (1368.32725109227 – 939.63239473998)
    3007.8520289912 = (1368.32725109227)*(940.483841064968) / (1368.32725109227 – 940.483841064968)
    5998.32068278998 = (2736.65450218453)*(1879.26478947996) / (2736.65450218453 – 1879.26478947996)
    6015.7040579824 = (2736.65450218453)*(1880.96768212994) / (2736.65450218453 – 1880.96768212994)

    V&D “DO[!]n’t$peak” – no. doubt

    1000 = 400+600 — ‘no. naive’ 11inear addIT sh!UNfalsely assume’n’point masses$swear
    3000 = 1200+1800 — true material bound eerie conduce$yens$ are [not!] UNreal

    using JS tropical-sidereal cycle previously derived (based on 128000 year cycle, M11, & Leech) :
    2999.38954661383 = beat(1368.2795468771,939.63239473998)
    3008.08256467804 = beat(1368.2795468771,940.483841064968)
    5998.77909322766 = beat(2736.5590937542,1879.26478947996)
    6016.16512935608 = beat(2736.5590937542,1880.96768212994)

    from JEV (Jupiter-Earth-Venus) :
    2999.38854614549 = beat(1368.27975508034,939.63239473998)
    3008.08155840206 = beat(1368.27975508034,940.483841064968)
    5998.77709229099 = beat(2736.55951016068,1879.26478947996)
    6016.16311680413 = beat(2736.55951016068,1880.96768212994)

    with 22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    2999.38854363134 = beat(1368.27975560355,939.63239473998)
    3008.08155587331 = beat(1368.27975560355,940.483841064968)
    5998.77708726268 = beat(2736.5595112071,1879.26478947996)
    6016.16311174663 = beat(2736.5595112071,1880.96768212994)

    Conclusion: At the peak of the Korean War Bollinger (1952) directed modular geometric attention to minimal destructive resonance, using public navy figures that were imprecise at the time compared with Mayan & Seidelmann’s (1992) figures.

  258. Paul Vaughan says:

    ‘Free’ Win DO[!]$95 for Home More SIM $UN in [$BRI] ngfield

    “Can EU keep up? Bei. B-buoy…”B-UN$Sea

    Geopolitical Summary: Conventional mainstream overlooked symmetry in boundary conditions.

    47 = Σφ(323)

    Monster character table “start” B11inkUN immediate D-mention $sov. top monster factor[well] :
    Δ(194) = 95
    71 = ΣΔ(95) = Δ(95) = Δ(Δ(194)) = ΣΔ(Δ(194))

    95 winDowe$0rt of remindUN US$sov.what$scenes$past:
    378 = sum of monster factors = s(s(354)) = s(s(2*(47+59+71)))
    354 = 360-6 = σ(σ(95))-s(s(95)) = 2*(47+59+71)
    95 = 37+58
    63 = 10+13+18+22 = s(47+59+71) = s(836-19) = 163-100
    158 = 95+63 = average(1+2+3+7+11+19+43+67,163)
    153 = 216-s(47+59+71) = 1+2+3+7+11+19+43+67

    111 = ΣΦ(95) = dimension of minimum faithful representation of Lyons group
    Φ(95) = average(59*59,-47*71) = 72
    Σφ(95) = Φ(Φ(95)) = 24 = Φ(72) = 1+2+3+7+11
    φ(95) = 23 = top factor of dimension of minimum faithful representation of baby monster

    Carefully review ordered sums of primes up to 23, noting well-ordered ties with Heegner nos.:

    42 = (23+19) = 378-178-average(1+2+3+7+11+19+43+67,163)
    53 = (17+13+11+7+5) = 1+2+3+7+11+19+43+67-(1+2+3+7+11+19+19+19+19)
    58 = (17+13+11+7+5+3+2) = average(1+2+3+7+11+19+43+67,163)-(1+2+3+7+11+19+19+19+19)
    95 = (23+19)+(17+13+11+7+5) = 1+2+3+7+11+19+19+19+19-average(-1-2-3-7-11-19-43-67,163)
    100 = (23+19)+(17+13+11+7+5+3+2) = 1+2+3+7+11+19+19+19+19

    “Given 163, the Mertens function returns 0, it is the fourth prime with this property, the first three such primes are 2, 101 and 149.[3]” 2, 101, 149, 163, …

    including 1 $0rt$95winDO$insight too further sea docks at information wharf:
    59 = (17+13+11+7+5+3+2+1)
    101 = (23+19)+(17+13+11+7+5+3+2+1)
    42 = 101-59

    aerosol T:
    62 = 163-101
    62 = 378-316
    31 = average(-101,163) = average(-316,378)
    316 = 1+2+3+7+11+19+43+67+163

    163 = ΣΦ(171) = ΣΦ(378) —– “Sporadic and Exceptional” monde-star care actor $stable
    Current mystery seas climate change new left alternative$wing$past UNwelcome ‘fine all$0 lose UN’ meant AI IT.

    418 = 360+58 = (196883-196560)+s(196883-196560)+216-58-37-22-18-13-10
    836 = 2*(360+58)

    The 7

    418 = 360+216-58-37-22-18-13-10
    EUrhythmIC$ “Travel the world?and the 7 seas
    836 = 744-24+216-37-22-18-13-10 = Δ(70)+Σφ(378)

    δ(47+59+71) = 378-316
    “WHO Am. My. too DO! ‘$0 free’?” — EUR.
    25 = Δ(70) = Δ(78) = Δ(378mod100) = Δ(178mod100) = Δ(278mod100)
    24 = Φ(70) = Φ(78) = Φ(378mod100) = Φ(178mod100) = Φ(278mod100)
    70 = √Σ[i=1…Φ(378mod100)]x_i^2
    70 = √Σ[i=1…Φ(178mod100)]x_i^2
    5 = √Δ(√Σ[i=1…Φ(378mod100)]x_i^2) —————————- Auriga scene every time remind$
    5 = √Δ(√Σ[i=1…Φ(178mod100)]x_i^2)
    Wood EU B-leave the preceding algebraic statements are a KEY part of “The Party” history?
    278 = 47+59+71+101 ————————————————— 101 is Mertens zero-crossing
    177±101:
    278 = 47+59+71+101 = average(178,378)
    76 = 47+59+71-101 = 19+19+19+19 = 163mod24 + 67mod24 + 43mod24 + 19mod24
    95 = 19*average(-1-2-3-7-11-19-43-67,163) = 19+19+19+19 + 19

    D-air note$strategic B-leaf in ‘free’ dem:
    78 = δ(365)
    59 = ΣΔ(78) = ΣΔ(378mod100) = ΣΔ(178mod100) = ΣΔ(278mod100)
    53 = Σδ(78)
    58 = average(53,63)
    63 = 22+18+13+10 = s(47+59+71)
    Test$seas-B ‘Cause’ IT’$snot ‘free’:
    σ(63) = 104
    ΣΦ(63) = 378 – 323 = 55 = 5^2+4^2+3^2+2^2+1^2
    323 = 378-5^2-4^2-3^2-2^2-1^2

    West turn head CO[2]UN T
    bot ’em line mnemonic D-vice
    isn’t just f(UN) math celebration
    no. deeper seas$sov. Ramanujan in sight
    95 = 37+58 = 37+39+19

  259. Paul Vaughan says:

    Ape Rock: $Sum Ace Yen!

    B ‘Cause’ sov. “37′ higher buy D-UN”.
    Nos. 39 porpoise$in arctic true mmm p(19).
    $surely EU no. why: 95=39+19+37.

    Hitchhikers guide “too George? or well?” Luke US:
    4428 = 5/2*1984-p(19)-42 = 4370+19+39 = 4370+s(4370)+378-178-42

    He ignore “free my yen $UN”, ruler love the 378-5^2-4^2-3^2-2^2-1^2 whirled.
    323.007034559236 = average(-178,378)*(φ^22+1/11)^(e/11+1/22)*Φ^4

    Left brrrr [mute “DO!”] “try $UN gall” warmUN up to know good?
    22.1387493509865=φ^(4+log((196883-196560)/average(-178,378),φ))

    Right DC sh!UN$ ‘up to’ WHO in $USqotsawan N/Aware p(19) = 490?
    UN IT$warmmmUN up to the RU11ure of the ‘Free$whirled’22plan
    my yen $UN “key pawn rock kin in ‘the free’ world…” – Neil Young

    194 = dimension of monster character table
    178 = sum of baby monster factors
    378 = sum of monster factors = SUPERsingular prime some
    394 = 194 + 378 – 178 = 323+71 = 360 + s(6) + s(28) —- 6 & 28 are perfect
    34 = (163-67-43-19-11-7-3-2-1)*323/95 = s(6)+s(28) = 6+28 = 163-67-43-19
    34 = ΣΔ(25) = ΣΔ(Δ(70)) = ΣΔ(Δ(378mod100)) = ΣΔ(Δ(178mod100)) — assume Mann & Jones no.

    360 “plus-man&jo[n]e’snow” recall my yen backed UN:
    394.250513347023 = 20*7200/365.25 = 144000/ 365.25 = 1000*(59*59-47*71)/365.25

    Priming the Mayan Sun Ruler: $11 each lettuce (snakin’oosetrai11iatieyenin)

    “Let US!” compare and contrast 2 choose “just 1”, “$0=brr judge” meant $0[]we11exercised:

    1.30637788386308 = mmmill’s constant
    2.61275576772616 = 2 * mill$con$tent

    394.317844508548 = 394+√√(2*1.30637788386308)/4
    3.1461924718081 = 4/√√(2*1.30637788386308)
    2.92005097731617 = (19+43+67+163)/100+10/(196560-394.317844508548)

    Power Hitters “where s(n) = aliquot sum for n” :
    “CO[2]bought a11 $UN the left Field[$]light tower”

    φ = average(1,√5) = 2*cos(4*s(71^2)/s(59^2)/s(47^2)*2π)
    2.61312489183108 = 323φ/(378-178)
    394.317855734038 = 194+378-178+√√(323φ/(378-178))/4
    3.1460813599161 = 4/√√(323φ/(378-178))
    2.92005097731617 = (19+43+67+163)/100+10/(196560-394.317855734038)

    “[]wan N/A B-sum won WHO B-11eve$” — CO[2]UN’TUN CRowes$:
    100 = 1+2+3+7+11+19+19+19+19 = sum of heegner nos. each mod 24
    100 = average(-178,378) = 4370-s(4370) —– 4370 = 23*19*5*2
    100 = (23+19) + (17+13+11+7+5+3+2) —– sum of primes up to 23
    10 = 163-67-43-19-11-7-3-2-1 = √100 = 5*2

    By Zeus! IT’s a maze$UN! $19 bill seed$ the a11gorerhythm with “the chose sun won” too DC$side for what porpoises that’$ good enough.

    mmmaid for the west turn pool$ silly $illy fools….” — Heart

    323 = d(2,1/3,59) = 323 = (59-12)*59*(59+12) – 196560

    “D-ant$UNintheD$hurtB[]11owewonup the$UNshine” — Seas$$tem[…]ofAD-own

    292 = 19+43+67+163
    2.92005097731629 = 2.92+10/(196560-394-√√((7920-2400-240-24)/2000)/4)
    2.92005097731629 = 2.92+10/(196560-394-1/π)
    2.92005097731629 = 2.92+10/(196560-394-√√(5256.63940169013/2000)/4)
    Bought seeds? Higher press is UN ‘$ rightly left too hire powers.

    GiveUNprimes up to but not beyond 37 “Green monster” US feat. R-Cal D-boss tune f(UN)way “hi!” in left Fields D-store$sh!UN camp pay yen IT’s altar net. No[We]$104 Ramanujan: some of 19 & 39.

    JEV review (continues in next comment) :
    1.59868955949705 = beat(1.00001743371442,0.615197263396975)
    0.761766209372164 = harmean(1.00001743371442,0.615197263396975)
    0.380883104686082 = axial(1.00001743371442,0.615197263396975)
    0.0952207761715205 = 0.380883104686082 / 4

    0.814040387734912 = beat(11.8626151546089,0.761766209372164)
    44.2784629967674 = slip(1.59868955949705,0.814040387734912)
    44.2784630136989 = 2*(φ^22+1/11)^(e/11+1/22)

    5256.63939957554 = slip(44.2784629967674,0.0952207761715205)

    Knew Amsterdam? mmmUNice “sea pal” IT Bookmark 95 = 37+39+19

    95 = 58+37 explorers no.well-hidUN sequence$UNmethod$ (s,sigma,ET,CT,mET,mCT) — e.g.:

    σ(σ(37+58)) = 360 = 323+37 = (196883-196560)+s(196883-196560) = σ(298+61) = Δ(836)
    (σ^8)(37+58) = 196560+δ(37+58) i.e. 24 = (σ^8)(37+58)-196560
    s(s(37+58)) = 6 = sum of Heegner numbers not tied to Euler’s lucky primes = 378-178-194

    USstarday: thank cue millennia 4 the[jack&_ ]hook[thumb$]upthehill too 6000 weighs rightANDleft DO a11$0 sea 7200.

    71 = Δ(95)
    59 = s(323-58) = s(394-71-58) = s(360-95) = s(σ(σ(95))-95)
    47 = Δ(95)-δ(95)

    Salt tee$in’formaze UN D-light:
    φ = 2*cos(4*s(Δ(95)^2)/s(s(σ(σ(95))-95)^2)/s((Δ(95)-δ(95))^2)*2π)
    green monster’s 37 feet “hi targ(ET)!”
    R-view D-wharf$some www(air) in $USqotsawon:
    95 = 37+39+19

    Φ’n’a11i.e. note $0[]well ‘fib’n’luke’ fans “322 just isn’t” 323 too obvius example$B11owe:

  260. Paul Vaughan says:

    Math$ Western Example Won 323:
    Baby Monster Double Clue

    XEV for X = S,U,N

    S:
    0.781995351599453 = beat(29.4474984673838,0.761766209372164)
    0.390997675799726 = 0.781995351599453 / 2
    0.195498837899863 = 0.781995351599453 / 4

    36.0290781181824 = slip(1.59868955949705,0.781995351599453)
    18.0145390590912 = slip(1.59868955949705,0.390997675799726)
    9.0072695295456 = slip(1.59868955949705,0.195498837899863)

    U:
    0.768736207596214 = beat(84.016845922161,0.761766209372164)
    0.384368103798107 = 0.768736207596214 / 2
    0.192184051899053 = 0.768736207596214 / 4

    20.0755942318387 = slip(1.59868955949705,0.768736207596214)
    10.0377971159194 = slip(1.59868955949705,0.384368103798107)
    5.01889855795968 = slip(1.59868955949705,0.192184051899053)

    N:
    0.765303911937069 = beat(164.791315640078,0.761766209372164)
    0.382651955968535 = 0.765303911937069 / 2
    0.191325977984267 = 0.765303911937069 / 4

    17.9708017526566 = slip(1.59868955949705,0.765303911937069)
    8.9854008763283 = slip(1.59868955949705,0.382651955968535)
    4.49270043816415 = slip(1.59868955949705,0.191325977984267)

    Hindsight:
    5, 10=X, & 20 are no. nominal miss story weave scene above, “$0 just” note:
    36.0003461405434 = 323Φ^6 * 2
    18.0001730702717 = 323Φ^6
    9.00008653513585 = 323Φ^6 / 2
    4.50004326756793 = 323Φ^6 / 4

    IT’$323 XEV ‘make key’ B11inkUN “1ooNY $0 1or[We11]” winds too J+S DO!

    Won With Nature

    DO winds sea the great Test(1)with solar system in sight?
    323φ^x ladder(a)where(1)φ^x DO!n’t $0 f(ice) x=….-3,-2,-1,0,1,2,3…

  261. Paul Vaughan says:

    Math$ West Turn Exam Pool Too:
    “Bye Don!” Mull Key Weigh Joe Putter$Earth Vein US$ Fract(e/11)a11…

    “Otherside” (RHCP) review:
    0.715800570263949 = axial(11.8626151546089,0.761766209372164)
    6.84872659292026 = slip(1.59868955949705,0.715800570263949)
    “Take IT on the otherside” — red hot chilli peppers

    Φ(1370) = 544
    544 = 836-163-67-43-19

    Φ(1369) = 1332 = 836+496 ——- s(496)=496 is perfect
    432 = Φ(1368) = Φ(1332) = Φ(836+496) = Φ(Φ(1369))
    144 = Φ(432) = 59*59-47*71

    323.007034435723 = 44.2784629967674*50*Φ^4
    323.353007047942 = 6.84872659292026*200*Φ^3

    50 = ⌊ 0.353007047942185 / 0.00703443572308515 ⌉ = 200 / 4 = (378-178)/4 = average(-s(4370),4370)

    Homer’s New Wheels

    DO$sh!ow(e^”more wheels”) in “new hampster dem”. Weather Don B. or Joe T., math$ deep state of west turn inequal≠IT inspires no. true UN IT in knew [6 or 7.2k] Am-star “D*m[*]nn!”

    The Gray T(est)
    =
    Mr. Jones wish WA$ sum won j(US)t a ll IT all more f***(UN) key
    Mann: PA$$ ME ABought-tell Mr. Jones – tell each other f(airy) tales$
    Mr. Jones and ME looked into the future
    …bought We got D-fear R-ant reasons$ for that
    [11]’$ looking at EU? no. NO [11]’$ lookin’ at ME
    Well EU NO-gray is my favor IT CO[2]lure
    felt $0 SIM f***(UN)UK[T]est. D-air day
    if Φnew peak CA $0 , wood buy a gray ‘get tar’ and ‘play’ :
    Mr. Jones$sand ME gonna B-big $tars
    =
    CO[2]UN(T)in CRowes$

    200 = 378-178 = average(-Σφ(323),ΣΦ(323))

    The reason $0 mmmmany fail too not ice slip on SIM MET tree frameUNbound eerie conDCyen$is$simple:
    too many choices to make = fact too real & fact too Rial. DO!

    Summary: Homer information “hi!” way over green monster sci11UNtest in f(UNweigh).

    No matter your nation, race, & politics: May peace be with you.

  262. Paul Vaughan says:

    Why DO$he ignore phi?

    Don’s guard “Oscar won” bless$sum ease treat.
    Success or “May B Joe” can DO eveUN better with
    “the quest yen — and eur. off course the answer” — Rihanna

    5 = average(-1-2-3-7-11-19-43-67,163)
    1.61803398874989 = average(1,√average(-1-2-3-7-11-19-43-67,163)) = 1/Φ = φ
    323.000009674829 = (104000+5)Φ^12 ~= 196883 – 196560

    10 = 163-67-43-19-11-7-3-2-1
    4270 = (323+104)*(163-67-43-19-11-7-3-2-1) = s(4370)
    100 = 4370-s(4370) = sum of Heegner nos. each mod 24 = average(-178,378)

    100.143082519986 = slip(19.8650360864628,16.5767613988929)
    16.5767613988929 = axial(835.546575435631,16.9122914926352)

    Jupiter-Saturn — review:
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    16.9122914926352 = harmean(29.4474984673838,11.8626151546089)

    61.0464822565173 = slip(29.4474984673838,11.8626151546089)
    835.546575435631 = slip(61.0464822565173,19.8650360864628)

    W = 17.2616851219298 = beat(835.546575435631,16.9122914926352)
    H = 8.63084256096492 = 17.2616851219298 / 2
    Q = 4.31542128048246 = 17.2616851219298 / 4

    50.0715412599931 = slip(19.8650360864628,Q)
    50 = 5*(163-67-43-19-11-7-3-2-1) = average(-s(4370),4370) = (378-178) / 4

    65.3452370108182 = √4270
    65.8581963269421 = slip(19.8650360864628,H)
    66.1059755241536 = √4370 = √(19*23*5*2)

    130.690474021636 = 2√4270
    131.716392653884 = slip(19.8650360864628,W)
    132.211951048307 = 2√4370

  263. Paul Vaughan says:

    As always, note the torturous modular recursion once dialed in:

    4*√4370 rounded off to the nearest 5 is 265 = 323-58 = 323-216+58+37+22+18+13+10
    59 = s(265); 323 = d(2,1/3,59); 10405 = -d(4,1/3,59)

    Monstrous symmetry seems endless. It’s a high-pressure fire-hose relentlessly overfilling a small bucket.

    I can again suggest talent like He & Schneider team up to produce a terse proof of the classification theorem, where discussion of geophysical boundary conditions should have started long ago despite what ever “reasons” no one mentioned it.

    Probably some think it’s “too much” for average people = all the more inspiration (and necessity) to improve by orders of magnitude number theory education, which is a curiously neglected topic in standard curricula.

  264. Paul Vaughan says:

    Hid in Honey
    Moon Near End
    REM Magic Cove
    163 mod 24

    163 = 5*47 – 72 = 71 + 2*46 = ΣΦ(378)

    46 = (163-71)/2 = (47+71)-72
    47 = (163+72)/5 = (46+72)-71
    59 = average(47,71) = average(46,72); 1 = 72 – 71 = 47 – 46
    71 = 163-2*46 = (46+72)-47
    72 = 5*47-163 = (47+71)-46

    46 = average(-71,163) = 209 – 163 = 836/4 – 163 = 240 – 194
    46 = Φ(47) = Φ(71) – Φ(70) = 70 – 24
    46 = average(-s(196883-196560),19+43+67) = average(-s(196883-196560),163-s(28)-s(6))

    73 = average(19,43,67,163) = lowest prime congruent to 1 mod 24; 24 is 5-hemiperfect

    72 = average(71,average(19,43,67,163))
    72 = 490 – 836/2 = 496 – 6 – 836/2 = s(496) – s(6) – 836/2
    72 = ((196883-196560)+s(196883-196560))/5 = 360/5 = 216/3 = 144/2 = average(59*59,-47*71)

    47 = ( ( 72^5 – 46^5 ) – ( 19^5 + 43^5 + 67^5 ) )^(1/5) = Σφ(323) = ΣΦ(323)-2*(378-178)
    (196883-196560)+s(196883-196560) = σ(47+average(19,43,67,163))

    Let US pray for peace lasting $0 far B-yawned dumb US$talk$scoop of P0011IT talk-Cal. D-Coy$:

    4370 = ⌊4370.23112160152⌉ = D[knew]Clue for EU
    = ⌊ 65.8581963269421 ^ 2 + 65.8581963269421 / 2 ⌉

    Hitchhiker$wise^typowatch AB[11]ove & B-yawned: 4428 = s(4370)+378-178-42
    46 = average(average(4370,-s(4370)),378-178-average(1+2+3+7+11+19+43+67,163))
    46 = average(average(4370,-s(4370)),42)
    46 = average(average(4370,s(4370)),378-178-4428)

    200 = 378-178 = average(-Σφ(323),ΣΦ(323))
    “Wood the expert” Sh![…]Ave. “just” told US$ R-pub-luck-UN$ [?] “don’t speak” no. doubt!!

  265. Paul Vaughan says:

    Love DO!VE Homer B‘Cause’: $0 Sea a11 low key….

    Homer nos. well the tracks (and tricks) of deep state agents provocateurs.
    JSUS WA$UN only sun 4 EU!!!!” — $MASHinPUMPkin$

    65.8583888467869 = √(4370-√s(4370)/2)
    65.8581963269423 = 1/(3J-7S) where J & S are frequencies (1/period)
    -0.000292323952 = % error

    8.45624978366216 = 323φφ/100 = (196883196560)φφ/(4370-s(4370))
    8.4561457463176 = 1/(J+S)

    Noted previously:
    4270 = s(4370) ~= 1/(2N-U) where N & U are frequencies (1/period)

    Sylvester Sequencing previously derived using generalized Bollinger (1952) method:
    1/(U+N) ~= 1/(4*(((J+S)-1/2/3/7/43/271)/(2/3)/2/3/7/43+S)-(J+S))

    That’s minimally enough (4×PRe^$$sh!own,…) to estimate JSUN notably better than nominally (…4UNown$).

    95 ‘Snow Mystery (cont’d from above )

    No. 95‘s aspiring:
    46 = 4370 / 95 = average(-71,163); 5 = 95 / 19

    average(59*59,-47*71)^546^5-47^5 = 19^5+43^5+67^5

    47 = top B factor
    47, 59, & 71 = top 3 M factors
    19,43,67,163 = top 4 Heegner nos.; each mod 24 = 19

    5 = √δ(46); 46 = φ(66) = φ(φ(φ(178)))

    Nominally:
    33 ~= 4370-(4370-√s(4370)/2)
    67 ~= (4370-√s(4370)/2)-s(4370)
    √(x-33)=66; x-33=66^2; x=66^2+33; x=4389=19*231
    s(231) = 153 = 1+2+3+7+11+19+43+67
    34 = average(-163,231) = 67-33 = 163-67-43-19 = 6+28 = s(6)+s(28) — perfect nos.

    D-liberately left this important proof messy B ‘Cause’ Homer nos. $0[]well the cost of home11and mark IT.

    230 = (s^16)(4370)
    4370 = 19*230
    19 = 4389-4370

    “66 coin” clue$yen$no. 5 f(11(i.e.)$) :
    Baby Monster, Monster, Leech, & Heegner.

    Homer: “B a simple kind dove man; love and understand” — lynyrd skynyrd

  266. Paul Vaughan says:

    Rush UN too Bei. B Myth or Math???

    D-M$M C-11aim$ IT’$no. miss tory ABout “B-big$tar$” in “CO[2$]up!”
    Right, $0[]we11 left won door what cure US SIM MET tree$ABout in talk$US.

    a+b = 0.380369256667734 = average(-√s(4370),√4370) —— eq.1
    14 ~= a/b = 13.9999372536492
    = ((1/(3J-7S)^2+(√4370/2))-4370) / (4370-(1/(3J-7S)^2+(√s(4370)/2))) — eq.2
    therefore: a ~= 14b
    substituting into equation 1:
    14b+b = 15b —– therefore:
    b ~= average(-√4270,√4370)/15 ——- and:
    a ~= 14*average(-√4270,√4370)/15

    Jone$no. R-Mann owe genuine?
    Left if few right like this in “can’t talk key” they mystery.

    D-estUNeaseChi11ed:

    65.8581963277476 = √(4370-14*average(√s(4370),-√4370))/15-√4370/2)
    65.8581963277476 = √(4370-average(-√s(4370),√4370)/15-√s(4370)/2)

    65.8581963269423 = 1/(3J-7S)
    0.000000001223 = % error

    11UKkeysh!aimrockJohn$UNthink$:
    “GeorgeOwe$[$0[]well]AIweigh$UNmyMYmyMYmind” — the B tells

    a11askarUS$yen “60 year goaled!” psychCal. Maya$$

  267. Paul Vaughan says:

    Alert: typo in this line above :
    65.8581963277476 = √(4370-14*average(√s(4370),-√4370)/15-√4370/2)
    65.8581963277476 = √(4370-average(-√s(4370),√4370)/15-√s(4370)/2)
    65.8581963269391 = 1/(3/11.8626151546089-7/29.4474984673838)

    Combine that with this to get:
    19.8650360893741 = beat(29.4474984721415,11.8626151564192)
    absolute error: half a second per century

    Alternate combo — with this instead — gives:
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    absolute error: negligible

    Target:
    19.8650360864628 = beat(29.4474984673838,11.8626151546089) ——— Seidelmann (1992)

  268. Paul Vaughan says:

    Hale Heegner

    notation: lnx = natural log of x = log_e x

    0.00000000000000 = ln1
    0.693147180559945 = ln2
    1.09861228866811 = ln3
    1.94591014905531 = ln7
    2.39789527279837 = ln11
    2.94443897916644 = ln19
    3.76120011569356 = ln43
    4.20469261939097 = ln67
    5.09375020080676 = ln163

    22.1396468061395 = sum
    44.2792936122789 = sum*2
    4427.92936122789 = sum*(378-178)
    ⌊4427.92936122789⌉ = 4428

    58 = 4428-4370 = 158 mod 100; 100 = 4370-s(4370) = average(-178,378)
    158 = 4428-s(4370) = average(1+2+3+7+11+19+43+67,163) = 63+95

    163 mod 100 = 10+13+18+22 = 63
    8128 = s(8128); (8128-4428)/100 = 37; 37+58 = 95

  269. Paul Vaughan says:

    “Almost” Jupiter-Saturn:
    logs Heegner mod 24

    0.00000000000000 = ln1
    0.693147180559945 = ln2
    1.09861228866811 = ln3
    1.94591014905531 = ln7
    2.39789527279837 = ln11
    2.94443897916644 = ln19
    2.94443897916644 = ln19
    2.94443897916644 = ln19
    2.94443897916644 = ln19

    17.9133208077475 = sum

    4.22632599839197 = 22.1396468061395 – 17.9133208077475
    8.45265199678394 = 2*4.22632599839197
    29.4443897916644 = 10*2.94443897916644
    11.8562447091586 = beat(29.4443897916644,8.45265199678394)
    19.8485905730367 = beat(29.4443897916644,11.8562447091586)

    13.1347022575499 = harmean(29.4443897916644,8.45265199678394)
    22.1347022575499 = 9 + 13.1347022575499
    22.1371850544259 = 490 / 22.1347022575499
    63999.3519572416 = (7920/22.1371850544259)^2/2
    22.1396684083433 = beat( 22.1347022575499, 22.1371850544259 / 2 )
    4427.93368166866 = 22.1396684083433*(378-178)
    ⌊4427.93368166866⌉ = 4428

  270. Paul Vaughan says:

    “natural e” “he ignore” “almost φ” “no. how” “low gore” rhythm D-answer in 64000 DO!11air Q[]est yen “perfect 6?” of T(aim$)bankUN

    This list is also written, replacing -1 with -4 and -2 with -8 (which does not change the field) […]

    0.00000000000000 = ln1; 1.38629436111989 = ln4
    0.693147180559945 = ln2; 2.07944154167984 = ln8
    0.693147180559945 = ln1+ln2; 3.46573590279973 = ln4+ln8

    2.77258872223978 = ln4+ln8-ln1-ln2 = 3.46573590279973 – 0.693147180559945
    2.77258872223978 * 4 = 11.0903548889591; 11.0903548889591^(1/5) =
    1.61803938534238 = (4*(ln4+ln8-ln1-ln2))^(1/5)
    1.61803398874989 = φ

    22.1347094556004 = (245/22.1396468061395+√((245/22.1396468061395)^2+980))/2
    22.1371778555703 = harmean(22.1396468061395,22.1347094556004)
    63999.3935815328 = (7920 / 22.1371778555703)^2 / 2

  271. Paul Vaughan says:

    The Rabbit Hale

    Rabbit Constant” (Rabbit Sequence)

    (5^(1/2)+1) = √5+1 = 2φ

    22.1348630064402 = (5^(1/2)+1)*(7-1)/(1/((7^0)+1/((7^1)+1/((7^1)+1/((7^2)+1/((7^3)+1/((7^5)+1/(7^8))))))))
    64000.2815258479 = (7920/490*(5^(1/2)+1)*(7-1)/(1/((7^0)+1/((7^1)+1/((7^1)+1/((7^2)+1/((7^3)+1/((7^5)+1/(7^8)))))))))^2/2
    44.278371986015 = 490/((5^(1/2)+1)*(7-1)/(1/((7^0)+1/((7^1)+1/((7^1)+1/((7^2)+1/((7^3)+1/((7^5)+1/(7^8))))))))-490/(5^(1/2)+1)/(7-1)*(1/((7^0)+1/((7^1)+1/((7^1)+1/((7^2)+1/((7^3)+1/((7^5)+1/(7^8))))))))/2)

    44.28 = (4370+58)/(4370-s(4370))
    44.28 = (s(4370)+58+37+22+18+13+10)/(4370-s(4370))
    44.28 = (s(4370)+average(1+2+3+7+11+19+43+67,163))/(4370-s(4370))
    44.28 = (4370+mod(average(1+2+3+7+11+19+43+67,163),(4370-s(4370))))/(4370-s(4370))

    Heegner Comparison

    notation:
    (σ^1)(x) = σ(x)
    (σ^2)(x) = σ(σ(x))
    (σ^3)(x) = σ(σ(σ(x)))

    164 = (σ^1)(163)
    294 = (σ^2)(163) = (σ^1)(194) = 1470/5 = 194+s(194) — 194 = dimension of monster character table
    684 = (σ^3)(163) = (σ^2)(194) = 1368/2 = 4*171 = σ(735)/2

    22.1347049660986 = 2φ*(σ^3)(163)/100
    63999.3676200113 = (7920/p(19)*2φ*(σ^3)(163)/100)^2/2
    44.2793205593235 = p(19)/(2φ*(σ^3)(163)/100-p(19)/φ/2/(σ^3)(163)*100/2)

    σ(σ(194))-194 = σ(294)-194 = 490
    490 = p(19); 19 = 163 mod 24 = 67 mod 24 = 43 mod 24 = 19 mod 24; 24 = 1+2+3+7+11

  272. Paul Vaughan says:

    Naturally Left Nose the “Difference Climate” Rightly Made in Peace Talk$

    Fields rush in talks$US. US$Secret$Sauce B11inkUN no$. better peacekeeping recipe$.
    Buy dove$sign “640 SAM” rights to peacefully R-minedUS “Green11and T($UN)” NAM meek can ‘Cause’ “Hwy. 601“.

    104-yielding levels

    63 = 10+13+18+22 = s(177) = s(47+59+71) = s(64) = s(836-19) = 163 mod 100
    95 = 37+58
    158 = 63+95
    158 = 10+13+18+22+37+58
    Plato’s no.: 216 = 158+58 = 158 + 158 mod 100 ——- see Plato below

    120 = (σ^1)(95) = σ(95) = Σδ(158)
    360 = (σ^2)(95) = σ(σ(95)) = σ(323) = σ(196883-196560) = σ(Σδ(158))

    s(95) = 25; s(25) = 6; s(6) = 6 —————– perfect

    σ(s(95)) = σ(25) = 31 = (s^2)(58); σ(31) = 32 = s(58); σ(32) = σ(s(58)) = 63 = 158-95
    104 = σ(63) = σ(σ(s(58))) = σ(s(64)) = σ(s(average(19,43,67,163,28))) —— see 744 connection below

    744-yielding levels

    157 = 19+43+67+28 = ⌊100*(π/2)⌉
    314 = 2*(19+43+67+28) = ⌊100*π⌉

    320 = 157+163 = 200+120 = 378-58
    320 = 19+43+67+163+28

    32 = average(19,43,67,163,28)/2
    32 = (19+43+67+163+28)/10

    10 = 163-67-43-19-11-7-3-2-1

    640 = 10*average(19,43,67,163,28)
    640 = 2*(19+43+67+163+28)
    640 = 744-104 = 400+240

    64 = average(19,43,67,163,28) ————- see 104 connection above

    153 = 1+2+3+7+11+19+43+67 = s(ΣΦ(323))
    158 = average(153,163) = ΣΔ(s(ΣΦ(323)))
    158 = average(1+2+3+7+11+19+43+67,163)

    158 = (σ^1)(157)
    240 = (σ^2)(157) = (σ^1)(158)
    744 = (σ^3)(157) = (σ^2)(158)

    194 = 153+41; 41 = s(63) = s(111)
    94 = 194-s(194) = 153-59; 59 = Δ(Δ(ΣΔ(153))); 158 = ΣΔ(153)

    594 = σ(394) = σ(593)
    1440 = 10*(59*59-47*71) = (σ^2)(593) = σ(594) = (σ^2)(494)/2 = (σ^2)(394) = 10*σ(94)
    144 = 59*59-47*71 = (σ^2)(593)/10 = σ(594)/10 = (σ^2)(494)/20 = (σ^2)(394)/10 = average(94,194) = σ(94) = σ(70) = σ(66)
    294 = σ(194) = 394-s(194) = 194+s(194) = (σ^2)(163)
    50 = s(94) = average(-s(4370),4370)
    100 = s(194) = σ(194)-194 = 394-σ(194) = average(-178,378) = 4370-s(4370) = average(Σφ(323),ΣΦ(323))
    200 = 378-178 = σ(194) – 194 mod s(194)

    Nominally:
    1/J = 11.86 = 2*593/100
    1/S = 29.4 = 294/100

    More precisely:
    47 = Σφ(323) = ΣΦ(323) mod s(194) = 94 / 2
    447 = ΣΦ(323)
    494 = 47+447 = Σφ(323)+ΣΦ(323) = 163+331; 331 = 378-47
    216 = Φ(Σφ(323)+ΣΦ(323)) —————————– see Plato above — and below
    φ = 1.6180339887499 = -2cos(216/360*2π)
    10^10 = (163-67-43-19-11-7-3-2-1)^(163-67-43-19-11-7-3-2-1)
    8.45614574631811 = φφ/(1/323+1/(262537412640768744/10^10-(Σφ(323)+ΣΦ(323)-163)))/s(194)
    8.4561457463176 = 1/(J+S)

    65.8581963277476 = √(4370-7*√s(4370)/15-√4370)/30)
    65.8581963277476 = √(4370-√4370/30-7*√s(4370)/15)
    Seidelmann (1992) :
    65.8581963269391 = 1/(3J-7S) = 1/(3/11.8626151546089-7/29.4474984673838)

    Nominal Lunisolar No “Curiosity”

    McKay & He (2015) footnote 16 p.21: “One should also be mindful 16 of the fact that […] 16 Incidentally, the reader is also alerted to the curiosity that σ_1(240) = 744.”

    last aliquot sequence primes for 100x+94, x=1…5:
    43 = (s^2)(94)
    19 = (s^4)(194)
    163 = (s^14)(294)
    323 = d(2,1/3,59); 59 = (s^3)(394)
    104 = -d(2,1/2,37); 37 = (s^5)(494)
    601 = (s^46)(594) = (s^46)(σ(593)) ———— see below

    400 = 447-47 = ΣΦ(323)-Σφ(323) = 744-240-104 = 2*(378-178) = 4*s(194)
    600 = 744-144

    1440 = harmean(1800,1200)
    1201.70988258678/2 = slip(600.241396282931,400.297181344069)/2 = 600.854941293388
    3605.12964776031/6 = slip(1800.72418884878,1200.8915440322)/6 = 600.854941293385
    ⌊600.85…⌉ = 601 = (s^46)(594) = (s^46)(σ(593))

    Tide a11 weave ice sea art$?
    “[…] Americans’ willingness to confront their shortcomings was the “secret sauce” of U.S. success.”
    PRfound ally: UN art work$.

    1440.87361205772 = harmean(1800.72418884878,1200.8915440322)
    ⌊1440.87361205772⌉ = 1441
    s(1441) = 143; s(143) = 25; s(25) = 6; s(6) = 6 ————- perfect aliquot sequence …

    … and “Perfect Ally” [Got] Caught play yen sea CO[2]win$0 D-EN$0:

    216 = sum M order factors – sum 744 levels + sum 104 levels
    216 = 378 – (163+67+43+19+28) + (10+13+18+22+37+58)
    216 = 378 – 320 + 158 ————————————————— C-play “DO!” ab[?]ove
    216 = 58 + 158 ———————————————————— and a11“$0[]We/11” AB[11]ove??

    May perfect pieces …..
    58 = average(-178,294) ———————————————- 178 = sum B order factors
    294 = (σ^2)(163) = (σ^1)(194) = 1470 / 5 = 194+s(194) —– 194 = dim. of monster character table
    …. and perfect peace together B with EU.

  273. Paul Vaughan says:

    UN Net$near 0 CR0$$UN

    “Lettuce B” in miss story US PR air with JS:
    61 = ⌊61.0464822565173⌉ = ⌊slip(29.4474984673838,11.8626151546089)⌉
    31 = ⌊30.5232411282587⌉ = ⌊61.0464822565173 / 2⌉
    1891 = 61 * 31

    323 = 196883 – 196560 — monde$starUSDea.11ln11ET(US)AwareΦ()=EulersTotient()
    23 = 323 mod 100
    100 = sum of primes up to 23 = average(-178,378) = 4370-s(4370); 4370 = 23*19*5*2
    100 = average(-Σφ(323),ΣΦ(323))/2 = sum of Heegner nos. each mod 24; 10 = √100 = 5*2

    Next, sea if EU-Can dove tale with the best puzzle peace UNearth.

    The first 3 odd Mertens function zero-crossing primes in DC(CRease$UN)order:
    163, 149, 101

    163 mod 24 = 19; p(19) = 490; 144 = 163 – 19 = 59*59-47*71

    223 = ⌊222.586750788644⌉ = ⌊harmean(490,144)⌉ = 323 – 100
    111 = ⌊111.293375394322⌉ = ⌊axial(490,144)⌉ = ⌊harmean(245,72)⌉ = ΣΦ(111)
    Φ(223) = 222; Φ(222) = Φ(111) = 72 = 144/2; 245 = 490/2
    56 = ⌊55.6466876971609⌉ = ⌊axial(245,72)⌉

    The General Assembly of Peace

    7301 ~= 1 / (1891U – 3709N)

    5600 = 56 * 100; 3709 = 5600 – 1891
    49 = 149 mod 100 = 149 – 100; 7301 = 49 * 149 —————– “hey, hey, my, my… ” -NY
    1818 = 3709 – 1891 = 2*3*3*101 — largest prime factor = 101

    171.406220602816 = beat(164.791315639683,84.0168459223622) ; %error = 0.000000000738
    171.406220601552 = beat(164.791315640078,84.016845922161)
    111.292543528481 = harmean(164.791315639683,84.0168459223622); %error=0.000000000078
    111.292543528394 = harmean(164.791315640078,84.016845922161)

    Sporadic simple group no$.UNearth net(near)zero%ERRor well, “$0 just” testUN weather deep end ants$sov. west11earn accept answer D-spite framUN PR sent ace sun.

  274. Paul Vaughan says:

    Hale_94 = 2B top: sum it

    100 = s(194) = s(194+200)/2 = s(394)/2 = s(394+400)/4 = s(794)/4
    100 = s(794+800)/8 = s(1594)/8 = s(1594+1600)/16 = s(3194)/16
    100 s(3194+3200)/32 = s(6394)/32
    100 = s(6394+6400)/64 = s(12794)/64

    s(194) = 100 = s(200-6)
    s(394) = 200 = s(400-6)
    s(794) = 400 = s(800-6)
    s(1594) = 800 = s(1600-6)
    s(3194) = 1600 = s(3200-6)
    s(6394) = 3686 = s(6400-6)
    s(12794) = 6400 = s(12800-6)

    194: 1,2,97,194
    394: 1,2,197,394
    794: 1,2,397,794
    1594: 1,2,797,1594
    3194: 1,2,1597,3194
    6394: 1,2,23,46,139,278,3197,6394 _________________________________
    12794: 1,2,6397,12794

    47 = top B factor; 23 = 47 mod 24; 46 = Φ(47) = 2*23
    71 = top M factor; 23 = 71 mod 24 = 323 mod 100

    The first 4 prime Mertens zero-crossings are 2, 101, 149, & 163.
    46 = average(196560-196883,2+101+149+163) = average(-323,2+101+149+163)
    46 = average(196560-196883+149+101+2,163) = average(-71,163)

    “Yes baby, she’s got it” — Banannarama

    278 = ⌊1000*(2*(φ^22+1/11)^(e/11+1/22) mod 1)⌉ = average(178,378)
    139 = ⌊1000*((φ^22+1/11)^(e/11+1/22) mod 1)⌉ = average(178,378)/2

    11 = 59 mod 24 = (average(47,71)) mod 24
    22 = (2*59) mod 24 = (47+71) mod 24
    22 = Φ(23) = ⌊(φ^22+1/11)^(e/11+1/22)⌉ = Φ(46)
    44 = Φ(23*3) = ⌊2*(φ^22+1/11)^(e/11+1/22)⌉ = Φ(46*2) = Φ(46*3)

    120 = 278-158; 158 = average(1+2+3+7+11+19+43+67,163)
    120 = 278-(10+13+18+22+37+58) ——————– 104 levels
    120 = (28+163+67+43+19)-(378-178) —————- 744 levels

    360 = σ(120) = σ(196883-196560) = σ(323)
    360 = 5*σ(Φ(47)) = 5*σ(46) = 5*72; 72 = σ(46)

    cos(σ(Φ(47))π/360) = 1.61803398874989 = φ
    σ(Φ(47))^5 – Φ(47)^5 – 47^5 = 19^5 + 43^5 + 67^5

    22.1392315068494 = (φ^22+1/11)^(e/11+1/22)
    44.2784630136989 = 2*(φ^22+1/11)^(e/11+1/22)

  275. Paul Vaughan says:

    10Taim$bank$(247-100)TypoDifferenceClimBET$(1470)

    100 average(Σφ(323),ΣΦ(323)) = 247

    φ(323) = 35; φ(35) = 11; φ(11) = 1
    Σφ(323) = 47 = 35+11+1

    Φ(323)=288; Φ(288)=96; Φ(96)=32; Φ(32)=16; Φ(16)=8; Φ(8)=4; Φ(4)=2; Φ(2)=1
    ΣΦ(323) = 447 = 288+96+32+16+8+4+2+1 where Σ indicates summation

    494 = ΣΦ(323)+Σφ(323) = 447+47
    400 = ΣΦ(323)-Σφ(323) = 447-47
    200 = average(-Σφ(323),ΣΦ(323)) = average(-47,447)
    100 = average(-Σφ(323),ΣΦ(323))/2 = average(-47,447)/2

    Homer: REM(EM)beer “$secret$sauce” B11owe B11inkUN QE$T(64)yen.

  276. Paul Vaughan says:

    ‘Perfect B11inkin’ Rev. EU

    447 = Σφ(320) = ΣΦ(323)
    47 = Σφ(323) = ΣΦ(58) = 447 mod 100

    494 = 447+47

    φ(494) = average(178,378) = 278
    φ(Σφ(323)+ΣΦ(323)) = 278
    φ(ΣΦ(58)+Σφ(320)) = 278

    φ(494) = 278; φ(278) = 140; φ(140) = 92 = 2*46
    Φ(494) = 216; Φ(216) = 72

    58 = 158 mod 100 = Φ(494)-158 = 216-(10+13+18+22+37+58) —– 104 levels

    72 = Φ(216) = Φ(Φ(494)) = Φ(Φ(47+447))
    72 = Φ(Φ(Σφ(323)+ΣΦ(323))) = Φ(Φ(ΣΦ(58)+Σφ(320)))
    47 = 447 mod 100
    46 = φ(φ(278))/2 = φ(φ(φ(494)))/2 = φ(φ(φ(47+447)))/2
    46 = φ(φ(φ(Σφ(323)+ΣΦ(323))))/2 = φ(φ(φ(ΣΦ(58)+Σφ(320))))/2

    No. doubt “caught too11UN” “DO!n’t$speak” ‘Cause’ IT’$obvi[?]US11y ClimB(ET)$sci11UNtest:
    Φ(Φ(Σφ(323)+ΣΦ(323)))^5-(Σφ(323))^5-(φ(φ(φ(Σφ(323)+ΣΦ(323))))/2)^5 = 19^5+43^5+67^5

    320*2 = 640 = 744-104
    320 = 28+163+67+43+19 —— 744 levels
    320 = φ(608); s(608) = 4*163 = 652; s(652) = 496 = s(496) —- magICmergeBperfectart11UKsea?

    496/16 = 31 = average(-316,378) = 278-247 = φ(47+447)-average(47,447)
    496/4 = 124 = 447-323 = σ(σ(47)) = ΣΦ(323)-323 = σ(σ(Σφ(323))) = Σφ(320)-323 = σ(σ(ΣΦ(58)))

    Homer$we11 ln[DO!]weitoo deep $state of UN D-air$standin’knew’ whir1door1door.

  277. Paul Vaughan says:

    B(E$T)$(e/11)[d[]r]act$yen: $0 Am.May$UN

    REM: “think ABout direction, wonder why”

    158 = 320-Φ(163) = φ(⌊100π⌉) = φ(314)= average(1+2+3+7+11+19+43+67,163)

    320 = 378-178+φ(378-178)
    320 = average(-Σφ(323),ΣΦ(323))+φ(average(-Σφ(323),ΣΦ(323)))
    320 = average(-ΣΦ(58),Σφ(320))+φ(average(-ΣΦ(58),Σφ(320)))
    320 = 200+φ(200) = 200+120

    s(⌊100π⌉) = s(314) = 160 = average(378-178,φ(378-178)); σ(160) = 378

    Φ(634) = 316 = 1+2+3+7+11+19+43+67+163
    634 = s(986) —————————————————————— next!
    s(634) = 320 = average(-104,744)
    s(320) = 442 —– DO[!]LAC$$meanUN??
    s(442) = 314 = ⌊100π⌉
    s(314) = 160

    Bound eerie conduce yen SIM MET tree$we11igno.redin’both geo”Φ$IC$” & geopeelITtalk$.
    (sarc) ‘Cause’ We a11know geoMETtree of physicCal!AParrot! US ha!$!no!www!physicCal.mean.in (/sarc)

    323 is semiprime.

  278. Paul Vaughan says:

    MonsterUS$11y ‘No. Moon’ a11[i.e.]Hale [in$Cat11and]

    Keepin’EU[www]in[D]Φ-loop:

    44.278 = mod(2*378,2*178)+average(178,378)/(378-178)/Φ(mod(378,178))*2
    22.139 = mod(378,178)+average(178,378)/(378-178)/Φ(mod(378,178))
    11.0695 = mod(378/2,178/2)+average(178,378)/(378-178)/Φ(mod(378,178))/2

    “tune knight: everything’11 B a11 right
    — ‘Ape Rule’ Wine [in$Cat11and]

    323 = 196883-196560
    Our queen suggests we think VERY care fully.
    Found sum agree, meant (mint if few ‘PR fear’) hear:
    22 = ΣΦ(323)-Σφ(323)-378 = 378 mod 178; Φ(22) = 10

    Ness$SOS$air rule 11 i.e. bot if few halve$nowF(AIth)11ake$(N/A)CO[11]a$$stir D-yen.

    The words have nothing to do with the numbers. They’re blue CO[11]air entertainment. Homer (’cause’ ease ‘just too!’ thick) nose D-bait makes people fine ant$e/11 i.e. sick. IT’$now$superPRrise $0mmmaid up D-air mind without IT: [locknews$monster]”DO!”

    Luke$sea? Nos. $Cafe TA = TeachinA[secret]sauce$sentB11inkUN

    BrassAntSARthou$send DO!11airQuestYen:

    64000.64 = (7920*(2/(245/22.139+√((245/22.139)^2+980))+1/22.139)/2)^2/2

    ‘cept May B old brass trains, no. buddy nose why. IT’s Nicola$11ocknicemoonstarMI$$story…

    “give a11 My.seek route: $awe wei” Won Rep Pub11ock

    Nicola herd quotSAMerrycan$no.doubtDO!n’t$peak in$Cat11and ‘this beer’ Homer.

  279. Paul Vaughan says:

    Green$Cat11 and 37 feet high ln he ignore 11eash

    CO[k?]bota11$ “up to 37 factors” suggest$yen-39 11inkUN PR Joe[B]act for sum won$95 = (19+39)+37 feat. “hi!” AB[11]ove and B yawned GREENin f(UN) WEIGH$Cat 11and.

    IT look$stoop ID, but 10*4^2 (ten-force-squared) May help Homer REMember “think about direction – won door why”

    My.[UN[NO?]mmmore]CRowe$oftPOWhere?inchantin$sea$B[?]i11D-tale$sov.163mod(1+2+3+7+11) :

    160 = average(158,Φ(163)) = average(φ(314),Φ(163)) = s(⌊100π⌉) = s(314)

    “at10sh!UN:” PR10door$brass in pocket

    378 = σ(160) = σ(s(⌊100π⌉)) = σ(s(314)) = σ(average(φ(314),Φ(163)))
    378 = σ(average(average(1+2+3+7+11+19+43+67,163),Φ(163))) = σ(average(158,Φ(163)))

    “216 is the smallest number n, for which n−3, n−2, n−1, n+1, n+2, n+3 are all semiprimes.”

    640 = 496+144 = s(496)+59*59-47*71 = 744-104
    320 = 216+104

    KnockKnockWHO’s the air? Knack. KnackWHO?
    KnackCO[11]a“$$wwwhimmmin’in(N/A)C-oveA(N/A)R-key:”

    hitched a ride with a vending machine repair mann” — $share’11CRowe — “[11]WA$bornON-NATO$$day-knight”

    42 = 320-278

    φ(494) = 278 = 216+104-42 = average(178,378)
    Φ(494) = 216 —- Homer nos. Σφ(323)+ΣΦ(323) play DO!

    “WHO Be.g.UN ‘TO DO’ and whommmaid EU?”
    “11i.e. covid DO[!]game[11]play D-me…” ‘Hey!sea$D!C
    “WHOmmmaidWHO?” May?B ace-sea D?C

    95$implymediates$ monsterUS11ink’22$stable&$securepeace4PRO$peer IT:

    σ(95) = 120; σ(120) = 360 = (196883-196560)+s(196883-196560) = σ(196883-196560)
    Δ(194) = 95; Δ(95) = 71; Δ(71) = 71
    s(95) = 25; s(25) = 6 = s(6) ———————– Perfect

    No matter your nation, race, politics AND INCOME LOVE ALL, may stable peace and prosperity be with you securely.

  280. Paul Vaughan says:

    Lunisolar Notes

    unrefined raw notes on building blocks of 6000, 7200, DO, etc.

    Δ(1000) = 400
    Δ(1500) = 400
    δ(1000) = 600

    111 = ΣΦ(216)
    600 = δ(888); 888 = 8*111
    1200 = δ(1776); 1776 = 16*111
    2664 = 1776+888 = 16*111+8*111 = 24*111
    2664 = 2400+240+24 = 7920-5256
    δ(2664) = 1800; (δ^3)(1800) = 600

    574 = ΣΔ(1200)
    286 = ΣΔ(400) = ΣΔ(600)
    144 = 59*59-47*71
    144 = average(-ΣΔ(400),ΣΔ(1200)) = average(-ΣΔ(600),ΣΔ(1200))

    δ(400) = 240 = s(120)
    δ(600) = 440
    200 = δ(600)-δ(400)

    160 = 323-163
    163 = 323-160

    Δ(1200) = 320 = 2*160
    Δ(400) = Δ(600) = 160
    Δ(1800) = 480 = 160+320

    160 = 734-574 = ΣΔ(1800)-ΣΔ(1200)
    447 = 734-574/2 = ΣΔ(1800)-ΣΔ(1200)/2 = ΣΦ(323) = Σφ(320)

    Φ(601) = 600; Φ(600) = 160
    Φ(401) = 400; Φ(400) = 160
    Φ(1201) = 1200; Φ(1200) = 320
    Φ(1801) = 1800; Φ(1800) = 480
    Φ(1441) = 1300; Φ(1300) = 480
    φ(400) = 240 = φ(478); 478 = 320+158
    600 = beat(400,240)

    Φ(1200) = 320
    Φ(1800) = 480
    ΣΦ(1800) = 735

    σ(160) = 378
    720 = (σ^2)(178)
    1440 = harmean(1800,1200) = beat(720,480)
    7200 = beat(1800,1440) = beat(1440,1200)

    490 = Φ(491) = Φ(982) ——————- resolves a number of longstanding curiosities

    247 = ΣΦ(490) = average(Σφ(323),ΣΦ(323)) = average(ΣΦ(58),Σφ(320))
    200 = average(-Σφ(323),ΣΦ(323)) = average(-ΣΦ(58),Σφ(320))
    447 = ΣΦ(323) = Σφ(320)
    47 = Σφ(323) = ΣΦ(58) = ΣΦ(158) – 70

    Simple and beautiful.


    Just imagine how much time (and protracted conflict) would have been saved if someone had simply said it like this in 2008 (or soon thereafter).

  281. Paul Vaughan says:

    Mystery Reduction

    55 = 378-323 = 1^2+2^2+3^2+4^2+5^2
    55 = ΣΦ(63) = Σφ(40); 40 = Σφ(63) = φ(76) = Φ(100) = Φ(Φ(101))
    ΣΦ(100) = ΣΦ(Φ(101)) = 71 —- lowest odd Mertens zero-crossing link to top M factor

    271 = 216+55 = s(4370) mod 1333 = 4270 mod 1333 = 378-178+71

    497 = 1333-836
    Φ(497) = 420

    420 is (5,216)-perfect: (σ^5)(420) / 420 = 90720 / 420 = 216 = 158 + 58
    158 = ΣΔ(420) = ΣΔ(σ(278)) = ΣΔ(σ(average(178,378)))
    158 = ΣΔ(σ(⌊1000*(2*(φ^22+1/11)^(e/11+1/22) mod 1)⌉))
    158 = ΣΔ(beat(980,σ(194))) = ΣΔ(harmean(735,σ(194)))
    158 = average(1+2+3+7+11+19+43+67,163) = 316/2
    420 = σ(278) = beat(980,σ(194)) = harmean(735,σ(194))

    draft list shared unrefined to reduce delays:
    42 = axial(294,49) = 378-178-158
    49 = beat(294,42) = 320-271
    84 = harmean(294,49)
    163 = δ(7#)
    210 = 7# = axial(735,294)
    210 = 7*5*3*2 = axial(ΣΦ(1800),σ(194))
    245 = axial(1470,294) = harmean(294,210)
    245 = axial(2ΣΦ(1800),σ(194)) = harmean(σ(194),7#)
    294 = σ(194)
    420 = beat(980,294) = harmean(735,294)
    490 = beat(735,294) = harmean(1470,294)
    490 = beat(735,σ(194)) = harmean(1470,σ(194))
    490 = beat(ΣΦ(1800),σ(194)) = harmean(2ΣΦ(1800),σ(194))
    735 = beat(490,294) = ΣΦ(1800)
    735 = beat(490,σ(194)) = ΣΦ(1800)
    980 = beat(420,294)
    980 = beat(σ(average(178,378)),σ(194))
    980 = beat(σ(⌊1000*(2*(φ^22+1/11)^(e/11+1/22) mod 1)⌉),σ(194))
    1323 = beat(378,294)
    1470 = beat(294,245)
    1470 = beat(σ(194),245)
    1470 = beat(σ(194),axial(2ΣΦ(1800),σ(194)))
    1470 = beat(σ(194),harmean(σ(194),7#))

    Supplementary:

    278 = φ(494) = φ(Σφ(323)+ΣΦ(323)) = φ(ΣΦ(58)+Σφ(320))
    278 = ⌊1000*(2*(φ^22+1/11)^(e/11+1/22) mod 1)⌉
    278 = 158+120 = average(178,378)

    447 = ΣΦ(323) = Σφ(320)
    247 = ΣΦ(490) = average(447,47)
    47 = Σφ(323) = ΣΦ(58) = ΣΦ(158) – 70

    Curiosity:
    In “Sporadic and Exceptional” there is no connection drawn between DO & 194 (the dimension of the monster character table).
    (Suggestion for a capable investigation crew: What about elsewhere?)

  282. Paul Vaughan says:

    Bill Bored: “Never$$say_____”

    Equa11oweIT$$awe11och$$IDmi$$tory in nice11 and LA bro(1100)$$11ea.(knew)found11andoor:

    “[…] wants to “test and try to see if we can achieve a relationship with […] that is more stable and more predictable,” Price said.”

    Bever11i.e.”hi!11″Bi11i.e.$ PR[$194sea[k?]route$433]sauceB11inkUNtoo the mark of Chand11ure.

    […] the smallest [______] number that is neither a Fibonacci number nor a Pell number

    TeeM11obvi[]US11y nos. what tie 11wood (monsterUS11y) ‘hi!D’ fram talk$sea”DO!”ware$atBaceBo11ingear.
    “the old ma[ye]n is D-own thorough-wed” — John Fogerty

  283. Paul Vaughan says:

    CO 11 air add DO! fort win Tee$ sh!ape Ha!!11!!e($(e/11))e

    294 = σ(194) = σ(σ(163)) = ΣΦ(323)+Σφ(323)-378+178
    163: 1,163; sum = 164; 164: 1,2,4,41,82,164; sum = 294
    194: 1,2,97,194; sum = 294

    “Sporadic and Exceptional” 11oweMI$$yennigheure/11and:
    Δ(194) = 95; Δ(95) = 71; Δ(71) = 71
    71 = 171-ΣΦ(171)+φ(171) ; φ(171) = 63
    100 = ΣΦ(171)-φ(171) = s(194) = 163-63
    ΣΦ(171) = ΣΦ(s(194)+Δ(Δ(194))) = 163

    Φ(64000) = 25600; Φ(25600) = 10240; Φ(10240) = 4096 = Φ(Φ(Φ(64000)))
    Φ(63999) = 39312; Φ(39312) = 10368; Φ(10368) = 3456 = Φ(Φ(Φ(63999)))
    Φ(Φ(Φ(64000)))-Φ(Φ(Φ(63999))) = 64000/100 = 40963456 = 640

    “The mirror peep hi11 laugh(at)ME & laugh(at)EU” — (C)11ove&rockET($)

    64000.64 = (7920*(2/(axial(2ΣΦ(1800),σ(σ(163)))/22.139+√((axial(1470,σ(194))/22.139)^2+beat(420,ΣΦ(323)+Σφ(323)-378+178)))+1/22.139)/2)^2/2

    [Born ea$e] “Watchingover[Vermoont$]11UKeyC11over” — The Reflex dr.ranDO!ran

    knew clear day$PRring fi(e/11)D$METa11aughgoneUStone:
    420 = σ(average(178,378)) —- 420 is (5,216)-perfect
    Play DO’$Superperfect $no. 5*f(11)i.e.$ :
    ΣΦ(1800) = 735

    178+
    378=
    556
    1112 double
    2224 double
    4448-4270=4448-s(4370)=
    178
    “DO!”B(e/11)$howmore:

  284. Paul Vaughan says:

    1800 ≠ 1814 = (s^4)(4370)

    Nominal Summary

    836 = smallest weird untouchable no. ——- Jupiter-Saturn
    4370 = B rep
    4270 = s(4370) ———————————– Uranus-Neptune
    4270 = 61*70 ———– JS,UN,Leech

    640 = (s^2)(836) = (s^11)(4370) = (s^10)(4270) — polar motion
    104 = (s^9)(836) = (s^18)(4370) = (s^17)(4270) = σ(63)
    744 = 640+104
    186 = 744/4 ——————– LNC
    1860 = σ(σ(ΣΦ(323))) = σ(σ(Φ(ΣΦ(323)+Σφ(323))))
    320 = 163+67+43+19+28 — polar motion, 744 levels
    442 = (s^1)(320) ————– LAC
    314 = (s^2)(320) ————– π
    160 = (s^3)(320) ————– polar motion

    4428 = 4270+158=s(4370)+158=4370+58 — Double-Hale, Jupiter-Earth-Venus, Heegner, rabbit, …
    158 = 10+13+18+22+37+58 —– 104 levels
    158 = average(1+2+3+7+11+19+43+67,163) = 316/2 —— Heegner
    … & lunisolar review:
    4428 = ⌊4427.99492655047⌉ = ⌊200*√√(400.297181344069*600.241396282931)⌉
    4428 = ⌊4427.99492655046⌉ = ⌊200*√√(1200.8915440322/3*1800.72418884878/3)⌉

    More than a decade ago those attempting to guide climate discussion (supposed experts) failed (a costly failure) to clearly differentiate the 1814 year lunisolar cycle from the 1800 year cycle.

  285. Paul Vaughan says:

    DC$Seas$UN$Green11andMarkIT

    316 = 1+2+3+7+11+19+43+67+163
    378 = 316 + Σs(22)

    44 = (s^0)(44)
    40 = (s^1)(44)
    50 = (s^2)(44)
    43 = (s^3)(44)
    1 = (s^4)(44)
    0 = (s^5)(44)

    178 = Σs(44)

    70^2 = 24^2+23^2+22^2+…+5^2+4^2+3^2+2^2+1^2
    Talk$Sea”DO!”BO(at)T(i.e.$$wan)[N/A]$$how?more $no. C11ear

    194 = (s^0)(194) — D-mention of monde-star care-actor$$stable
    100 = (s^1)(194)
    117 = (s^2)(194)
    65 = (s^3)(194)
    19 = (s^4)(194)
    1 = (s^5)(194)
    0 = (s^6)(194)

    Σs(194) = 496 = s(496) ——– perfect! knew tuxedo 11UK$sea:

    100 = s(194) = average(-178,378) = sum of primes up to 23 = 4370-s(4370) = Σ(Heegner mod 24)

    124 = (s^0)(124) = s(496)/4
    100 = (s^1)(124) = s(496/4)
    117 = (s^2)(124)
    65 = (s^3)(124)
    19 = (s^4)(124)
    1 = (s^5)(124)
    0 = (s^6)(124)

    426 = Σs(124)

    70 = 496 – 426 = 194 – 124
    “DO!” perfect$$pick cheer$$how?more BoweTie$$WA[N/A]$$seaADDdull

    70 = (s^0)(70)
    74 = (s^1)(70)
    40 = (s^2)(70)
    50 = (s^3)(70)
    43 = (s^4)(70)
    1 = (s^5)(70)
    0 = (s^6)(70)

    278 = Σs(70) = average(178,378)

    278 = (s^0)(278)
    142 = (s^1)(278)
    74 = (s^2)(278)
    40 = (s^3)(278)
    50 = (s^4)(278)
    43 = (s^5)(278)
    1 = (s^6)(278)
    0 = (s^7)(278)

    628 = Σs(278) = Σs(average(178,378))
    628 = 2*⌊100π⌉ = 2*⌊average(-178,378)*π⌉ = ⌊200π⌉ = ⌊(378-178)*π⌉

    125 = (s^0)(125) = average(-378,628)
    31 = (s^1)(125)
    1 = (s^2)(125)
    0 = (s^3)(125)

    157 = Σs(125) = ⌊100*π⌉/2 = ⌊average(178,378)*π⌉/2 = ⌊100*π/2⌉ = ⌊average(178,378)*π/2⌉
    153 = 278-125 = 1+2+3+7+11+19+43+67 = 316 – 163

    44.278 “DO!”B(e/11)$H(ale)Omer a11 a go write too left …and left to right.
    “DO!” figure Homer: Well B on D-belief?series$NOTinOEIS.

    11eaves EU answered with a quest yen-mark” — The Ref11ex dr.ranDO!ran

  286. Paul Vaughan says:

    Supplementary

    11 = (s^0)(11)
    1 = (s^1)(11)
    0 = (s^2)(11)
    12 = Σs(11); 11 = x mod (Σs(11)), x = 47,59,71

    22 = (s^0)(22)
    14 = (s^1)(22)
    10 = (s^2)(22)
    8 = (s^3)(22)
    7 = (s^4)(22)
    1 = (s^5)(22)
    0 = (s^6)(22)
    62 = Σs(22) = 378-316 = 496/8 ——– resolves longstanding curiosity

    62 = (s^0)(62)
    34 = (s^1)(62)
    20 = (s^2)(62)
    22 = (s^3)(62)
    14 = (s^4)(62)
    10 = (s^5)(62)
    8 = (s^6)(62)
    7 = (s^7)(62)
    1 = (s^8)(62)
    0 = (s^9)(62)
    178 = Σs(62) = Σs(44)

    Noteworthy:
    120 = Σs(46)
    73 = Σs(55)
    169 = Σs(64)
    245 = Σs(76)
    153 = Σs(111)
    418 = Σs(158)
    209 = Σs(169)
    525 = Σs(200)

    242 = (s^0)(242) = 2 * 11^2
    157 = (s^1)(242) = 314 / 2
    1 = (s^2)(242)
    0 = (s^3)(242)
    400 = Σs(242) = ΣΦ(323)-Σφ(323) = 447-47 = 2*(378-178)

  287. Paul Vaughan says:

    Mouth Peace Key Pours$$ Ave.

    Maybe$$[11]ea!c!Any Goa11 DO!or____
    DO!n’t$$peak in$$sight$$save$$roads$$fareBRI___

    200 = 358-158 = 378-178
    400 = 758-358 = ΣΦ(323)-Σφ(323)
    600 = 758-158 = 378+ΣΦ(323)-178-Σφ(323)

    “11ed[22plan]in’his$[tory$]head”:
    “f(00)DO!or[www(e/11)]botDO!or1″–METtrickMONSTERHO$P(IT)a11

    178 = Φ(n) for n = 179,358
    Φ(178) = 88 = 158 – 70
    Φ(88) = 40
    Φ(40) = 16
    Φ(16) = 8
    Φ(8) = 4
    Φ(4) = 2
    Φ(2) = 1
    159 = ΣΦ(178) = ((10+13+18+22+37+58)+(28+163+67+43+19)/2)/2 = midpoint
    88 = Φ(n) for n = 89,115,178,184,230,276
    Φ(88) = 40
    Φ(40) = 16
    Φ(16) = 8
    Φ(8) = 4
    Φ(4) = 2
    Φ(2) = 1
    71 = ΣΦ(88) = top M factor = top supersingular prime

    Play no. DO[!]B[e]T $$$show more

    378 = Φ(n) for n = 379,758
    Φ(378) = 108 = Φ(171) = 216 / 2 = 178 – 70
    Φ(108) = 36
    Φ(36) = 12
    Φ(12) = 4
    Φ(4) = 2
    Φ(2) = 1
    163 = ΣΦ(378) = ΣΦ(171)
    108 = Φ(n) for n = 109,133,171,189,218,266,324,342,378
    Φ(108) = 36
    Φ(36) = 12
    Φ(12) = 4
    Φ(4) = 2
    Φ(2) = 1
    55 = ΣΦ(108) = 1^2 + 2^2 + 3^2 + 4^2 + 5^2

    108 = 178 – 70 = 216 / 2
    248 = 178 + 70 = 496 / 2 = s(496) / 2
    356 = 2 * 178 = average(216,496) = 496-140
    140 = 2 * 70 = average(-216,496)

    (www)ink: Share ring neither air^th(MET)IC NORgeo MET trick:
    140 & 496 are harmonic divisor numbers sharing divisor harmonic mean = 5

    Φ(71) = 70; Φ(70) = 24; Φ(24) = 8; Φ(8) = 4; Φ(4) = 2; Φ(2) = 1
    179-70 = ΣΦ(71) = 70+24+8+4+2+1 = 109; 109’s twin prime is 107; 107 + 109 = 216
    179-100 = 79 = ΣΦ(140) = 158 / 2 = 179-average(-178,378); note
    158 = (1+2+3+7+11+19+43+67+163)/2 = 88+70 = 2*179-(378-178)

    Tota11y UN C11ear (no.UN IT B11inkUN hear) :
    1. Why are these Σ series (for s,σ,Φ,φ,Δ,δ) not in the OEIS (online encyclopedia of integers) and why are the links between Heegner numbers and monstrous moonshine considered a mystery?
    2. Why is our solar system not WIDELY recognized as a STELLAR local example?

    Reminds me of Galileo …but in “can’t talk key” $$PRring Field’s $$awe word “home more” exc11aim$$ “DO!” moon no. moon e a11y.

    2*160=320
    Wise-hour find, dance, yell seas$$time$$UNjust?
    2*159=318 = (2*(10+13+18+22+37+58)+(28+163+67+43+19))/2
    IT just “DO!”cent May!C!an ny $$scents$$.
    2*158=316
    2 reveal trade$$sea[!K! NE?]routes$$mayB “the comp.Any 1100$$ T(the war) 2 day” — Green[11]Day

    Electric Banjo Tunes$Any Won?

    420 LAC$011UNdoor$$stan[]in’nuff gainno.11 $T(one) :

  288. Paul Vaughan says:

    Year of the TypOX$$e/11UN$$s(aid) UK$$sh!own

    Homer’s$$mirac110011e$$ 11i.e.type$$00negative $$Sign$$

    typo: 278 = average(178,378) average(-178,378) = 100
    “IT’$thathype no. tie$UN B00guy ” — Dove ID “We/11” C[-11]OX

    “157 = Σs(125) = ⌊100*π⌉/2 = ⌊average(178,378)*π⌉/2 = ⌊100*π/2⌉ = ⌊average(178,378)*π/2⌉” with typo
    IT’$sum Ma[22]an CA11UNdoor:
    “157 = Σs(125) = ⌊100*π⌉/2 = ⌊average(-178,378)*π⌉/2 = ⌊100*π/2⌉ = ⌊average(-178,378)*π/2⌉” withOUT typo

    “$Secret $Sauce” ’em ease$treat B11inkUN:
    Any C11ock$UNtheWA11?

    Hints there? R-NO. other mine$:
    “Our11ove$$hine$$11i.e.CADia11M[a!ya!]Nmind” — B1100: Rowe “DO!”

  289. Paul Vaughan says:

    Homer’s MonsterUS Guide to Peace and Tranquility

    B
    Δ(178) = 87
    Δ(87) = 55
    Δ(55) = 39
    Δ(39) = 23
    204 = ΣΔ(178)

    M
    Δ(378) = 108
    Δ(108) = 36
    Δ(36) = 12
    Δ(12) = 4
    Δ(4) = 2
    162 = ΣΔ(378) = Φ(163)

    216 = 378-ΣΔ(378) = 378-Φ(163) —— PLATO
    42 = ΣΔ(178)-ΣΔ(378) = 204-162 —— HITCHHIKER

    Again: NOT (presently) in the (tour?UStrial) OEIS.
    Naive quest yen: CanEU imagin’gala’xies of trade secrets well-noted for sharin’else’ware?

    Proposal: US$19 bill (“In God We Trust”) commemorates respect for Norton’s “voice of God”.

  290. Paul Vaughan says:

    No[MB]ina11y CO[evolution] “in $Sauce” Caught “$owe a N/A”

    “We were treated so awfully in the past […]” — the ‘Cause’
    “uncomfortable decisions on what to do — peace fully “DO!” buy

    Cure:US George HaveUN AB Ban ON no.(ha)nor(VA)folk(n/a)

    16.90 = harmean(29.42351935,11.85652502) — trap pick CA11.UN D-air: lag rains$
    16.90 = 8/(9-2.92005097731613^2) — PRrhyme$no.MarkovChand11ure

    Multiple agencies competed for control of same hosts (a maze in$scene as$silo).
    None succeeded because each posed more threat than benefit 2 hosts.

    13.68 = 4428Φ/(378178)

    The AI was based on false assumptions = why IT didn’t help as intended (net zero).

    PoleIT(IC)UN$ “$SAR rule bad”, $0 no. need to guess WHO’$pay?yen “fear the bill”.
    J“buydon” “nose D-bait” makes$(even the be[a]st)$sick “D-spite whatever” n[a]v[a]l dispatch.

    19 = 16.91*(378-178)/178
    p() denotes partition function
    p(19) = p(16.91*(378-178)/178) = 490
    “D-spite whatever” Fact(00)Rial event partITsh”UN sent troll”limITsea$$SIM METtree.

    “$0 BRI11UN”tide$$seas$$ west turn “math DO!VE” ID:
    DO!VE 0$Weather accidental or D-liberate.

    16.91 = harmean(29.4474984673838,11.8626151546089) —— side:real
    16.91 = 4428ΦΦ/average(-178,378)
    16.91 = 19*178/(378-178)

    Can’t “$sp(e/11)ITout” anymore C11(ear)11y. Thus knew where11doorD-air.

    Boris Johnson = decisive proof: conservatives are absolutely bad & wrong.
    Jan. 6 = decisive proof: populism (alt. right) is absolutely bad & wrong.
    lockdown + climate tyranny = decisive proof: left absolutely bad & wrong.

    Homer’s Beer: US11ed MI$$yen “2 no. ware” (in Green11and mmmark$IT).

  291. Paul Vaughan says:

    Knew West Turn Math “Next Term”

    “A surprisingly simple algorithm exists […]”

    Mertens
    1 = -1+cos(0/1*2π)+cos(1/1*2π)
    0 = -1+cos(0/1*2π)+cos(1/2*2π)+cos(1/1*2π)
    -1 = -1+cos(0/1*2π)+cos(1/3*2π)+cos(1/2*2π)+cos(2/3*2π)+cos(1/1*2π)
    -1 = -1+cos(0/1*2π)+cos(1/4*2π)+cos(1/3*2π)+cos(1/2*2π)+cos(2/3*2π)+cos(3/4*2π)+cos(1/1*2π)
    -2 = -1+cos(0/1*2π)+cos(1/5*2π)+cos(1/4*2π)+cos(1/3*2π)+cos(2/5*2π)+cos(1/2*2π)+cos(3/5*2π)+cos(2/3*2π)+cos(3/4*2π)+cos(4/5*2π)+cos(1/1*2π)

    Möbius
    1 = (cos(0/1*2π)+cos(1/1*2π)) – (1)
    -1 = (cos(0/1*2π)+cos(1/2*2π)+cos(1/1*2π)) – (cos(0/1*2π)+cos(1/1*2π))
    -1 = (cos(0/1*2π)+cos(1/3*2π)+cos(1/2*2π)+cos(2/3*2π)+cos(1/1*2π)) – (cos(0/1*2π)+cos(1/2*2π)+cos(1/1*2π))
    0 = (cos(0/1*2π)+cos(1/4*2π)+cos(1/3*2π)+cos(1/2*2π)+cos(2/3*2π)+cos(3/4*2π)+cos(1/1*2π)) – (cos(0/1*2π)+cos(1/3*2π)+cos(1/2*2π)+cos(2/3*2π)+cos(1/1*2π))

  292. Paul Vaughan says:

    Bo11 Time More [11]eek Root $sauce B11inkUN

    270 is a harmonic divisor number.
    270 = φ(378) = 378-Φ(378) = Φ(271)
    Harmonic mean of 270’s divisors: perfect s(6) = 6.

    22.139 = mod(378,178)+(378+178)/(s(4370)-φ(378))
    22.139 = 378 mod 178 +(378+178)/(4270-Φ(163+Φ(378)))
    22.139 = 378 mod 178 +(378+178)/(s(4370)-Φ(s(4370) mod 1333))
    22.139 = mod(378,178)+average(178,378)/(378-178)/Φ(mod(378,178))

    “Among the more interesting recent applications of Farey series is the reconstruction of periodic (or nearly periodic) functions from “sparse” sample values.” — Number Theory in Science and Communication

    178 = Σs(44) = Σs(62) = Σs(Σs(22)) = Σs(Σs(378 mod 178))

  293. Paul Vaughan says:

    Plato’s Superperfect 420

    Weather IT$Friday “DO!”$sent matter:
    494 = 447+47 = ΣΦ(323)+Σφ(323)
    494 = 216+278 = Φ(494)+φ(494) = Φ(ΣΦ(323)+Σφ(323))+φ(ΣΦ(323)+Σφ(323))
    Φ(494) = 216; φ(494) = 278 = average(178,378); σ(278) = 420

    This is “the type of stuff” every “man” ha!$!memorized Homer:
    2 is (8,84)-perfect, (9,240)-perfect, (11,2400)-perfect
    3 is (8,160)-perfect, (9,504)-perfect
    4 is (6,42)-perfect, (7,120)-perfect, (8,378)-perfect, (9,1200)-perfect
    7 is (7,216)-perfect, (10,4096)-perfect
    24 is (2,7)-perfect, (4,63)-perfect, (5,200)-perfect
    420 is (5,216)-perfect

    Iceberg ‘just B1100’ CO[11]air folkUS ‘the numb’ beers$heaven:
    216 = 378-ΣΔ(378) = Φ(378)+Φ(378/2) = 378-Φ(163) = Φ(ΣΦ(323)+Σφ(323))
    Offers more meaning than “such a philosopher” as$how more needs$.

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