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.

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.

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.

“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] ”

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.

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.

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..

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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 10at 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).

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 196883 – 196560 = 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.

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)

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.

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.

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.

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.

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

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:

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.
=

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)

=
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

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 [___]”

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.

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

“√(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

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

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.

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

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. wonheaR
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.

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.

“√(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”

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

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

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.

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.

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.

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.”

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.

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).

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.)

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.

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.

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.

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.

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).

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

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.

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:

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

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.

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.

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.

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.

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

Rightly We leftIT 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.

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).

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 perfectin 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

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”.

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.

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

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:

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”

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.

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.

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.

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.

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.

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-parte:
2*δ(Φ(59))*s(Φ(59)) = 1984

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

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

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:

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.)

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)”

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.
=

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 offactorized order call ’em.

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.

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

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.

“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

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.

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 life2011 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 workwas 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.

Only a few pages but worth a read IMO.

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..

Reblogged this on Climate Collections.

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.

Irretrievable: 2 comments vanished.

Political commentary on the “60 year linear component” of climate variables — motivated

not by climateso far as I can tell but maybe fear that western financial sanctions won’t have a valid military guarantor by 2030 or 2036 — ignoresthe more general contextof 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 strongparticipation turn-off and IT’s at least a cause ofmajor“behavior modification” (ware ITmay beinverted 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 surethey won’t pass the filter.Harry would no doubt be left with unanswered questions about the block of numbers and IT’s creative decode.

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

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.

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.

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.

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.

Eccentricity and apsides…

ECCENTRICITY EXCITATION AND APSIDAL RESONANCE CAPTURE IN THE

PLANETARY SYSTEM u ANDROMEDAE [2002]

This paper lays the groundwork for understanding theorigin 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’

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.

6441/3 ~= 4270/2

4270 * 13/12 = ~4626

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

=

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.— Queens of the Stone AgeThe walls are closing in again: O[r]well.”

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 relationsneedn’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

silentlymakingincorrectorganizational changes (downscale fractal ofmisledbig tech,no. doubt).At the

endof the first link change the example3to5.Round off the result to 4. That gives new input 4^5 for the

beginningof the second link.Repeat with

10at the right endof the first link.The result rounds off

too 2so 2^10 goes.in to the far leftof 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 redirectwhat curry US IT to196883 – 196560 =as fall lows:32325*√(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 * 323Half 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 folksnumb brrrsright down to street-levelaware 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 AImostX[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 TopsBAliquot sum (sum of proper divisors) for 2*5*19*23 = 4

370:4

270 = 1+2+5+10+19+23+38+46+95+115+190+230+437+874+2185“Velvet Ears”

123Flash mnemonICBack.in Hazard Count: TieWan M$y$n D-Po11ace171.406545820013 = 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) )Study with care: easy to derive

system at IC ally(it’s only a fewtrivialsteps).You’ll recognize not only 2^(1/J/2) & φ^4*2^(1+1/J/2) but exactly why they’re off and by

exactlyhow 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 SolBy part

tie sunIC topknewtrail’ski leftM 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

…distinctfrom 208.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).

Hi! K.in.on Boris Bluff

Too daze note U-N.Doors.cores

perfectnon.11.in.ear.IT.Hears the backs tory: dec.

aids a gosum BRIteechairs D-sidedschool-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.in

2400 reviewspill28IT /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.

Freedom is

Nota Dr.illBallparks can

BBo ring but knot fin weighin’ best tune where EU can seethe playbook writin’ on“the green monster” weather leftandright: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) = 224.991673680083836 / 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

sureyou 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.“Spam” in the C[ENSO]Rship

Can mod

phi nichefish file tour?104 ~= ⌊(e^√s(

298+25)π)^(1/2)⌉^2 – e^√s(298+25)π104 ~= ⌊(e^√s(

196883-196560)π)^(1/2)⌉^2 – e^√s(196883-196560)πmod 13

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.

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

cleandiscretion.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

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

IT’s possible

no. thingget$11earnedbyethe rightUN tell afterUSelection.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 outice?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).”Link to List of Aliquot Sums

11.in.ear

think kingsware 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

Bwritten?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:— 11ed 22plan “R(amble)ln”The autumn moon lights my weigh“

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

s(28) = 28IT 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)Vague O-Port Tune IT AIn’t

Typo:“Otherperfectnumbers include 2, 5, 52, 88, & 96.” — correct shh!UNtouchABLEFor 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.

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 criteriamore generally.This insight eluded us for too long:

208 ~ = slip(slip(24.067904774739,19.8650360864628),19.8650360864628)/2Once I saw that I understood the

naturalrole 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 compatibleSimilarly 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.

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.

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

“…and IT’s whisper-D that soon if we a11 ca11 the tune”— 11ed 22plan4th comm.in simplification & clarification suggests diagnostic comparison

of Seidelmann’s (1992)

2432 = 19 * 2^7

with

2436 =

29*842436 =

58*42——— General√(Φ-φ)Ramanujan’s Guide to the 9*11*IX*XI = N.in.8.O.wonSimplifying

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))

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.

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=2,3,4,6,12Twice 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

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) =

4270s(4270) = 4658

s(4658) = 2794

s(2794) =

1814s(1814) = 910

s(910) =

1106s(1106) = 814

s(814) =

554s(554) = 280

s(280) = 440

————————– see below

s(440) =

640s(640) = 890

s(890) = 730

s(730) = 602

s(602) = 454

s(454) = 230

s(230) = 202

s(202) =

104s(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) ——- perfect652/4 = 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

Branchde Rop fits into the aliquot sequence for the baby monster and 836 is

a dead-end branch— what it means to beuntouchable—off that.Arithmetic Dynamics

=

Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. […] A

fundamentalgoal is to describe arithmetic properties in terms ofunderlying geometric structures.=

For an illustrative example, take a look at how simple

thisrecipe is:“[…] “Mathematics may not be ready for such problems.” […] “is an extraordinarily difficult problem, completely out of reach of present day mathematics.”“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

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

PerfectFit19=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 ICguides?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)

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

/2/3/7/43s(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 Laskar4200 = 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

Rome and k

new join general….each let

us groupmonde stir us slystableparameter√(Φ-φ)shh!UN:d(

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

Incident

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

Standby 4 perfect primorial leech tie.in.

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”— “morehue moonthan WHO man” whyIT22oN?bei3 = √(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)

37 FEAT HI.

in Leftcaught sum green monster:Fieldsφ niche-D “a11y”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. whenso dense.Don’t Mayan D-proof: “just”

go ONwith Bei=lief.in 2+2=5 no. when4*s(496) = 1984is 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 paintedgreen 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^√37πBei. U-N D!Superst[r]ing

Reverse-eng.in.ear.

ring oldschool“luminary” assembly with 2020 hindsight, we findattached aperfectleech:490000 =

15*70^2/ asin(φ/2) * 2πLol! What curry US IT didn’t even

(“don’t speak”tell us. Tech no. CR at IC govern antno.doubt)s?ware.inmonde stir US $ sly “green”.Sylvester’s Sequence

(“more huegoesmoonthan WHO man” = whyITzoombei)as far as possibleaccount.in 4 minor deviations from superpastthe perfect leach,string 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— Lady G “just” D-antsB“Anchoring

thus contrastsoldschool “green”leach attachedto mainstream mindset verseUSSylvester’sdoubleexponential 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 strategypreciselyexposing an exponential inclearconflict with adoubleexponential. A hair-splittingsubtledifference in this caseis period-doubling.Scene IC Route to Secure IT

CountSol4370 = 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-antsΦ(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 KnowsReign D-a√(Φ-φ)r5106 = 4270 + 836 = 2*3*23*37

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) = 1584Φ(

1985) = 1584double:

Φ(3382) = 1584

Φ(3970) = 1584

highest:

Φ(6210) = 1584

Having

boldly lefta 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) = 1584Φ(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 exponentialconstraint with no local adjustment• global order

subtly-retuned (prime-D-iffewprefer)4fit withdouble-exponential local greed104.000034332275 = ⌊(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 thoselevels 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 mind

set to orientin rich territory.Conventional linear “thinking” about what’s “best” doesn’t help see (sort

and classify) howeverything 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 ofseverefinancial 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— “don’t speak” no. doubt√(Φ-φ) no. ya reel good”

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.

CorrectS[hh!]U

N[8]O[R]well.in.sightThe tech “know”

so-calledCR at IC “eXpeRts”neveradvise-D of period-dub e ln SIM MET try.“the bloop ill O-pens euRise

is theRe a bet te[a]

R weigh?a

knewReligion pRess scRibedto those with

outthe faithis IT 2 late 2 go back?

is IT 2 late 2 go?

theRe’s No.

wonheaRand peep e/11 eveRy waRe:

yeaR .oN euR

O-won” — Queens of the StoneAgeWon step back.

73500 = 3*(196883-4270/5-29)/8

2 steps

4word.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)/80.000004508890 = % error

Recall: s(4370) = 4270

First-order model’s base-

D on the baby monster.Leech

preciselyIDsthe pointof symmetry that’swell-correctedby Sylvester’sdouble-exponential greed.We’ll takethe seen ICroute awayfrom luck D-own weather Joe Trump or Don Biden.1 = 1/2 + 1/2 be 4+after USelection:

knowmatter what curry US IT obstructs, free dem. fromeur. increasingly savage financial terror schemesdirected 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 withno consolationround.PC’s plan’s UNstable. Withdrawn BE support can only be restored under more favorable circumstances should they

stablydevelop tosustaintrust — including: no more Boris $ under mind “free” doom for a monarchyplanninga reign ofwidespreadhomeland financial torment withno consolation.Ramanujan’s the man

who knewin.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

Can’s e/11-D: mmmIC

Dawn a11-D’s PerfectHire Arch IC AINotice225 = 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 aweEU’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))IC a few obviOus typos: IT’s

no. thing per sun AIon the C[ENSO]Rship.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 century1.61803398874989 = 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)

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

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

anyonnome?194 = average(225,163)

172 = 194-22 = 9+163

171 = 9+ET(163)

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

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

Notation AlertI’m refining notation — going back to R() for Ramanujan and

d() is the differenceof 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.

Kentucky Δoor B

Miranda Cheng:

“Moonshine is a puzzle that might require inspirations from everywhere and knowledge from everywhere.”key word Schneider

M-ETgolden mobius totientalt: https://arxiv.org/vc/arxiv/papers/1109/1109.3216v1.pdf

Kentucky Δoor B Classic

Miranda Cheng:

“Moonshine is a puzzle that might require inspirations from everywhere and knowledge from everywhere.”key word Schneider

M-ETgolden mobius totientJuncture 162With 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’ssimplyconsider: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 wellenough.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*163gives 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

HeegnernumbersΦ(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 exampleto 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 fansnote well where and howpieces fit the whole. We can see de Vries in discrete context without which de Vries mnemonICally wood knotB 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. onC[ENSO]Rship (weather easy or not) will hinge on “JunkSure” 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 onan otherwise hazard UStrail.Here I leave 1 mysterious note:

Φ(4374) = 1458

Won Course Braiding Streams from Left and Right

This comment will be

leftdeliberately 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 nontotientΦ(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 = 104IDs del pezzo surface structure.A del Pezzo

surf ace circuitstarts 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

ZZ$ox: Sharp Dressed Mayan$Top

Note well Heegner

M-ETdel 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 AgeMy[UN]key orient

superstring 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.R— Queens of the Stone Age “Monsters.eD.in he’er”inthe Paris Sol.”Ally Caught Up in Bushy Taled Top O Log IC AI Totient Trees

So far weave no.

comm.on.tory.onthe 3rdroot, which first sprouts59trees 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 algebrastorm.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.

Route

to Fifth Root.in Sight5StepsN8O worry O[r] left.

in on dawns golfrighter keys:“V(ic)tory— Putt, tee $myth “The War e ^ O[r] ”i$ Mayan“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 findscomm. on!ground.That was the

look forward to a simple ending.Now let’s

backtrack.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

simplyexplore fifth-root connect sun withsquare-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]nSomething te/11s

US this isn’twhat climb ET “super” comm. pew daresore luckkin fear.“fallOmysteryOjung11chi11ed…“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 branchshare ring aliquot sea stems(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.inphi 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

haledEU close.in victoryI don’t want

too tameEur. animal styleEU won’t be caged.in the call of the will.ed” — Patty Smyth “The Warrior”Red do

or wellnote won mystery66 left unadduressed 4 coverage UN[he!ll]O-there day with $O asin(O[r]We’ll).Hale 240

2, 12, &

240areboth highly and sparselytotient.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)^

ee-Luke Shh(Own)

Guard Do[w]nsPerfect Links!Sol Look! Shone27, 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—EUneed tuneO“THseasurfacetemperatch[ET]O[r]wellB(est) Tune Perfect Green CO2shh!UN Monster Level 7 in f(U-N weigh)

φ(8128) =

4096φ(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=2,3,4,6,12Again 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.”— GaussLeft field creatures: Nature’s beauties there to sea (surf ace stem peer ashore) if Eur. will.int(el)look

pastthe C[e^(NSO)]Rship. (IT AImost divided and conquered US.)Room O[r]s of

short circuitin.wash.un.tuneMayan D-Palace readers remember SUM thing CO2shh!UN reversing

the tale endof 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 endswith 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 worddoor.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[E

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

5Δ(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.

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 structurethat 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 ICto 496● Most won’t be ready for

top-level Heegner perspective…with type O’s left.in to follytempt UN fact-chuckers dare UN 2 fixmajortypos 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 sequencebrokethe perfect solar system superstring symmetry with period doubling symmetry (more on U-N in the Jovianreference framesoon….including a 0% error).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)).

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

1/11/500/211500

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

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

generouscampaign for superior math education — anoble imperativefor civilizations facing challenges which cannotbe solved with politics.George Polya’s advice:

“Solve aDone:simplerproblem.”1728 = 2*Δ(Φ(59))*s(Φ(59))O-Bay AI lie CAUGHT

Aliquot

.inmoonlightsightstree-top baby mondestirrin’ minimalrepresentation.faithfulHunted by

U-No. they’re predatorialowl, thecute test rightfly 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 ——

anothermember 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” conΦ

DOn’t EU AIrede^no.?Doubly greedy Sylvester’s (double exponential) Sequence precisely marks period doubling symmetry from perfect jovian superstring symmetry. The subtle difference is

severalorders 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

crystalclear. 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.West Turn Security Makes

298 BIGInvestments in Better Math EducationHomeland innocents

learnedto crave unilateral freedom

simplybecauseon CR Ave.,

U-N ill at Orwellfree doom because.Multilateralism — a word with

rapidlyincreasing negative connotations —meansbad dictators “representing” several billion savagely bully millions of good folks inconsequentlyfaltering democracies, just like they do to their “own”.In 1984UN 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 ONo!Emerge.in-C-meet-UN secure

ET count sol: western citizensmight use mathto explore elements sov. climb ET ….ifthey 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) =

104s(652)=s(4*163)=496=s(496) is perfect

163-104=59 trade

perfectseekroutes indeed!Φ(Φ(59)) = Φ(58) = Φ(29) = 28 = s(28)

Just divide

4UNother 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 mazeUN guide dense IT’s AI a curerat must fear. ICWHO invited sci11UNs bullies todemonstate the USmull tee laud orwell“leadership” “change”.If I were Donald Biden or Joe Trump I would NOT debate but simply state:

1. Pandemic: I will

notlock you down. Without delay wewill stoplockdown 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 predatorssurelydo, availed opportunity.2. Climate:

Superiormath learning outcomes in homeland math education are a securityimperative. We dare not face themonstrouscost 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+42hitchhiker guides orwelltoojovi-invariantD-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 729Φ(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 ACEshh!U-N…” – TWSΦ(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

rightbe 4 the lord” — Jack BlackTour Key’s Perfect

Neat O-String.in the French Moonlight496 = 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

EUstand WITH ORWELL.MLB is

majorleague baseball— and monster leech babyif 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 a11primeRudolph: With well-muzzled math hawk key stick to 6.

We

st TurnPresident’s Hexagonal Rev YouInnocent shorthand for hexagonal

is deliberately misinterpretedwithin Macron’smoonshoton 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(weave just numbers:“don’t speak”no. doubt)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/11O…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 = 31Why’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 onclimb 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

peacefullystop IT by whateverjustmeans as necessary.We started on this trail a few months ago. SUDDENLY it became necessary to SHARPLY alter course. Why? Rightly left a

totalmystery (totientif you prefer).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. facemsm fact-chuckersleft 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-Pub

Lock-D[oorwell]F(UN)D-red$hawk$park$way over Eur.

rise mod e ln blue moonshine 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 EUmoonlightblew: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?Hexagonalright 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 stirf(right).in leftFields RushMacron nom IC mmm ET a11: “I will choose free w[oo]l”.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.Nos. Reign

D-air Leftas a Mist EerieTie 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]onesHale O-We’UN Climb ET

Zipped PastSci11UNtestPerfect 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]lwoolover 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 betPR pro g,h, and i names will sun for borg rOw(e/1)1 to board of fly yens$quarrel climb ETdirect tears(“don’t speak” no. doubt).Adjacent

totient pairs differenceandcototient pairs sumtothe 3rdset 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 anotherpage in the sci11UNs moonlight.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 hereplus or minus 5 from 59 mean.in J[S]UN very ant1728 = (59-5)*(59+5)/2. See cure roots PET no. greenmonde stirs in the paris owlon Eur. sense soar shh!!B partitions 42 to define U:

4270=s(4370)

4370=2*3*

19*2342=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)incounterbalUNs.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.

Suddenly4*496 superstrings 1728+298-42no. mysteryin 1984. Left ITshelved right upUNtil notefar abovepoint-D to 4270 & 1584 from 4370.Last spring the missing link was math taxonomy keyed

nowhere nearbiology cataloguing standards to “solve a simpler problem” left right hearin’ “home” land D-vision.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.

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

exactlyat 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.

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.

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

“Start Again”

“There is no. thing

fairin this world.”836 = 1333-497

497 = 1333-836

“There is no. thing

safein this world.”2432 = 2*1216 = 4*608

“….and there’s no. thing

surein 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

sumthingleft.in this world. ” — Billy IDo[rwel]lGreen 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

Link 1

σ() = divisor sum

https://oeis.org/A000203

https://oeis.org/A000203/b000203.txt

s() = aliquot sum

https://oeis.org/A001065

https://oeis.org/A001065/b001065.txt

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)

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

Remember:

652= 4*163This stuff is stunning. It’s such a pleasure to work on it and see all the connections.

TheO[rwell]Air

s:IDweightWhat 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-parte:2*δ(Φ(59))*s(Φ(59)) = 1984

“I can still recall the time: wait!..TRIED to breakONthrew to THE

O[]airsID” — The DOORSPress 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 UNoffdawns a portforge[]O2 seajust in.history’s traitshow door.“Hey Joe, ware

EU go!inwith that … .. …. …. ?” — j.handrocksThe DOORSequence

“the gate is

straitdeepand 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*

271Φ(7#)/2=1+2+3+7+11=24

43-24=19

43+24=67

Sylvester coined the term “totient”.

163=

271-108162 = 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!..

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

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

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|=2711806=|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

preciselyin 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||

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.

=

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

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

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

Review

Why phi?

Ford Circles Sines Pythagorean Triangles + supplementary Ford Pythagorean

Ford Circles Fibonacci Lucas Phi Root 5

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 = 32*

3-1 = 52*

7-1 = 132*

43-1 = 852*

1807-1 = 3613Pythagorean spiral: a(n-1), a(n)-1 and a(n) are sides of a right triangle. 3, 5, 13, 85, 3613, 6526885,

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

IT Ski Rev EU: “Win September” 1984

Joe thank cue: Constitution of

1806 left infile truth tourpress “I don’t tell”[B]lock-D from 2020 piece search. West turn-(D Clock)-back 36years.Monster US Moon Ch!nes Won

and UN[all]liePure[“too Heegner” PR Joe ActD]fect L!nkinLet 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 1perfect 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 weatherprofoundly simple or(well)simply PR(O)f(O)UN-D()Near-tune herd “the voiceof God”.111=Φ(323)-(47+59+71)=Φ(196883-196560)-(47+59+71)

Ly : 163=2+3+5+7+11+31+37+67

“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

Count Sol Clear

Ants OrWellLuck D-own human writes “e

XpeRt” 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)^5x = 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 PRINT (IT SIM f**** O!N) UK” — LG“— the doorsbreakfast ware the news is red“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

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 reviewoffactorized order call ’em.