Orbital resonance and the celestial origins of Earth’s climatic changes – Why Phi?

Posted: October 30, 2021 by tallbloke in Analysis, Astrophysics, Celestial Mechanics, climate, COP26, Cycles, Ice ages, modelling, moon, Natural Variation, Phi, research, Solar physics, solar system dynamics
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A year after I wrote the original ‘Why Phi’ post explaining my discovery of the Fibonacci sequence links between solar system orbits and planetary synodic periods here at the Talkshop in 2013, my time and effort got diverted into politics. The majority of ongoing research into this important topic has been furthered by my co-blogger Stuart ‘Oldbrew’ Graham. Over the last eight years he has published many articles here using the ‘Why Phi’ tag looking at various subsystems of planetary and solar interaction periodicities, resonances, and their relationships with well known climatic periodicities such as the De Vries, Hallstatt, Hale and Jose cycles, as well as exoplanetary systems exhibiting the same Fibonacci-resonant arrangements.

Recently, Stuart contacted me with news of a major breakthrough in his investigations. In the space of a few hours spent making his calculator hot, major pieces of the giant jigsaw had all come together and brought ‘the big picture’ into focus. In fact, so much progress has been made that we’re not going to try to put it all into a single post. Instead, we’ll provide an overview here, and follow it up with further articles getting into greater detail.

One of the longest known climatic periods is the ~413,000 year cycle in the eccentricity of Earth’s orbit. This period has been found in various types of core sample data and discussed in many paleoclimatic science papers, along with cyclicities around 95, 112 and 124kyr, and shorter periods such as Earth’s obliquity variation, ~41Kyr and Earth’s equinoctial-precession periods of ~19 and ~23kyr. Stuart has discovered how all of these periods are related to each other and to the planetary orbits and their synodic conjunctions.

We’ve also been able to link these Earth Orientation Parameters and climatic periodicities to the planetary orbital and synodic conjunction periods which we believe are key to modulating solar activity. The basis for these were laid out in my 2011 post on Jupiter and Saturn’s motion and further developed with the valuable input of many Talkshop contributors, culminating in the solar variation models published by Rick Salvador and Ian Wilson in the 2013 special issue of Pattern Recognition in Physics.

Solar Total Solar Irradiance (TSI) prediction model hindcast created by Rick Salvador using planetary periods discussed at the Talkshop in 2013

Figure 1 below scratches the surface of what we have discovered. These relationships are all precise whole number ratios, not approximations. The red ‘Graham Cycle’ is a novel addition to previously known cyclic periods which connects the three areas of the figure; Solar-Planetary at the top, climatic periods bottom left, and Earth Orientation Parameters bottom right. Of note, are the ratios between the 60kyr Graham Cycle period and the periods in the three groups. They are mostly ratios of Fibonacci numbers or combinations of them. We know from a previous investigation that Fibonacci and phi (Golden Section) related periodicities tend to be stable and minimally resonant. It could be that the reason the 60kyr period hasn’t been found previously is due to it not showing up strongly in periodograms and other spectral analyses. Nonetheless, it’s an important period for our ‘Why Phi’ investigation and has a lot more connections than we wanted to clutter up Figure 1 with, as it already looks pretty busy!

Figure 1. Spatio-temporal diagram showing solar system dynamical arrangement with particular reference to solar modulation, climatic periods and Earth Orientation Parameters affecting Milankovitch cycles.

Solar cycles

Starting with the upper ‘Solar planetary’ section of figure 1, Ian Wilson’s 2013 PRP paper noted that the Hale cycle and Jupiter-Saturn synodic (J-S) have a 193 year beat period, which is evident in Oxygen18 isotope data as well as Group Sunspot Numbers and 10Be ice core data. This was picked up by the Helmholtz Institute research lab and covered in our earlier post on the Solar Magnetic cycle. What they didn’t pick up on is the fact that the same 193year beat period can also be derived from the 178.8yr Jose cycle and the 2403yr Solar Inertial Motion (SIM) period.

This second route to the 193 year solar magnetic cycle is a novel result revealed in this post. Using the beat period formula of (A*B)/(A-B) = period, the solar inertial motion cycle (A) proposed by Charvatova of ~2403 tropical years and the Jose cycle (B) produces the same 193 year result. It was then possible to tie all this together in the 60 kyr cycle shown in the diagram.

There are 336 Jose and 25 SIM in 60 kyr which means the beat period produces 336-25 = 311 solar magnetic cycles of 193 years each. The number of Hale cycles in 60 kyr is given by the number of J-S minus the number of solar magnetic cycles. i.e. 3024-311 = 2713. It’s notable that 311 and 2713 are both prime numbers. Coupled with the fact that the number of J-S in 60Kyr is the Fibonacci multiple 144×21, we think this is a strong indicator that both 193yr and 60kyr periods are significant solar-planetary cyclic periods.

Support for the 60kyr period comes from Russia, where in 2017 A. S. Perminov and E. D. Kuznetsov produced a paper at at Ural Federal University, Yekaterinburg, entitled ‘Orbital Evolution of the Sun–Jupiter–Saturn–Uranus–Neptune Four-Planet System on Long-Time Scales’. This paper shows inter-related variations in the orbital parameters of the gas giants including antiphase changes in the eccentricities and orbital inclinations of Jupiter and Saturn at ~60kyr and in-phase changes in those parameters at ~400kyr, antiphase to Uranus. These ~400kyr variations are likely to be drivers of Earth’s 413kyr eccentricity cycle.

ISSN 0038-0946, Solar System Research, 2018, Vol. 52, No. 3, pp. 241–259. © Pleiades Publishing, Inc., 2018.
Original Russian Text © A.S. Perminov, E.D. Kuznetsov, 2018, published in Astronomicheskii Vestnik, 2018, Vol. 52, No. 3, pp. 239–259
.

Planetary-climatic cycles

Moving on to the lower left ‘climatic and planetary cycles’ section of Figure 1,

The de Vries cycle is half of 21 J-S and is a prominent climatic cycle. It also links to other cycles through resonant harmonics: Hallstatt = 11 de Vries, J-S synodic precession cycle = 12 de Vries. 6 de Vries is 7 Jose cycles. 33 de Vries is 7 Eddy cycles. See also Why Phi? – Jupiter, Saturn and the de Vries cycle.

The lunar-terrestrial year (L-T) is 13 lunar months. Earth’s tropical year is used throughout this post. Whole numbers of both occur at 353 tropical years and 363 lunar years, forming 10 beats (363-353) of 35.3 years. An important period is 13 L-T, which is 2 Hallstatts and 11 de Vries cycle pairs (22 de Vries). This is 1/9th of the obliquity cycle. It is also 3x7x11 J-S. It follows that the 41kyr obliquity cycle is 3x7x11 Jose cycles, because the Jose cycle is 9 J-S. 3,7 and 11 are all Lucas numbers. We will post a separate article on the inter-relation of the Fibonacci and Lucas series, as they relate to orbital resonance. See also Sidorenkov and the lunar or tidal year (2016)

An explanation for the effect of the motion of the gas giants on these and other climatic periods is found in Nicola Scafetta’s 2020 paper ‘Solar Oscillations and the Orbital Invariant Inequalities of the Solar System’ discussed here at the talkshop.

EOP

At the lower right of Figure 1 we find Earth orientation parameters and associated cycles. To understand how these link to planetary periods we need to look at the motions of Jupiter and Saturn in particular. Kepler gives us this useful graphic in his book De Stella Nova (1606).

Kepler’s trigon showing the ~60 year cycle in the longitude of the Jupiter-Saturn synodic conjunctions. This may be linked to the ~60yr cycle of Earth’s major oceans, giving rise to the 30 year global cooling scare 1960-90 and the subsequent global warming scare 1990-2020.

From an earlier post: ‘As successive great conjunctions occur nearly 120° apart, their appearances form a triangular pattern. In a series every fourth conjunction returns after some 59.8 years to the vicinity of the first. These returns are observed to be shifted by some 7–8°’. Wikipedia. [2019 version]. After 3 J-S the conjunctions have nearly described an exact triangle, but the start position has moved (precessed) slightly, by 60/7 degrees of precession of the J-S conjunction axis. It takes 42 of those (42*3 J-S) to complete the precession cycle in 2503 years. (41×61.051 y = 41×360 degrees movement of the axis).

The 413kyr eccentricity cycle is equivalent to 55*3 of these J-S synodic precession periods, and 6765 or 55×123 (Fibonacci and Lucas numbers) of the 61.051 360 degree periods. Additionally 413 kyr = 10 obliquity periods.

In the brown triangle: the 19 kyr and 23 kyr periods have a beat period of the 112kyr perihelion precession.
23 kyr is 10 Hallstatt cycles.

In the blue triangle: the 95 kyr (5×19 kyr) and 124 kyr (3 obliquities) have a beat period of 413 kyr i.e. Earth’s eccentricity cycle (mentioned in various research papers). Since our 95 kyr = 353×270 and our 124 kyr = 353×351, we find: (351×270) / (351-270) = 1170, and 1170*353 = 413010 years (the obliquity period).

Discussion

The 95 and 124kyr eccentricity cycles are linked with glacial periods. From Park and Maarsch (1993) paper ‘Plio—Pleistocene time evolution of the 100-kyr cycle in marine paleoclimate records’: “The DSDP 607 time scale is more favorable to an abrupt jump in amplitude for the 95-kyr δ18O envelope, but not in the 124-kyr envelope. Rather, long-period δ18O fluctuations appear phase-locked with the 124-kyr eccentricity cycle some 300-400 kyr prior to its growth in amplitude and phase-lock with the 95-kyr eccentricity cycle in the late Pleistocene.” Because the 124kyr period is 3x41kyr (obliquity period), this may help explain the change from glacial periods around 41kyr to around 100kyr.

The bi-modality of glacial cycles and the 95 and 124kyr cycles is one of the modes of variation mirrored between celestial cyclic motion and Earth climatic events. There are also many periods which are ‘quasi-cyclic’ and vary in length within bounds whose attractor nodes fit our phi-Fibonacci scheme. We are not claiming to have elucidated a deterministic and predictable system with our precise whole-number orbitally resonant ratios. We are offering this scheme as a potentially useful roadmap for further investigations into the intriguing numerical links between planetary orbits, synodic timings, planar inclinations, eccentricities, energy transfers and other celestial mechanical and orientation data.

As an example of how our scheme links shorter to longer term cycles, there are exactly 9 Jupiter Saturn conjunctions in the period of the Jose cycle of 178.8 years. There are 55x21x2 Jose cycles in the 413kyr eccentricity period. Experienced researchers like Paul Vaughan will immediately see that this product of multiple Fibonacci numbers resolves to the product of the first 6 prime numbers 1,2,3,5,7,11.

The solar system is organised by the forces of gravity and electro-magnetism into a log-normal distribution of which the Fibonacci series and Lucas series are examples which maintain the stability of the system. Resonance is minimised, but also utilised to transfer energy between orbits in order to resolve inequalities through resonance-forced changes to the eccentricity and inclination of orbits. These changes give rise to the cyclic changes in climatic factors on Earth observed at all timescales from the ~22yr Hale and ~60yr J-S trigon to the ~100kyr and 413kyr glaciation in core sample data and other indices.

Data sources and acknowledgements

Planetary data used is from NASA JPL which gives the Seidelmann values for orbital periods. Our thanks to Paul Vaughan for insisting on their use.

The periods we have calculated can all be reproduced using the ratios we have provided on Figure 1 and the NASA JPL values for the Jupiter, Saturn and Uranus orbital periods.

Comments
  1. Paul Vaughan says:

    2317.99883398204 = 80*(√28+√163+√43+√19) ~= 2318
    4635.99766796409 = 160*(√28+√163+√43+√19) ~= 2318 * 2
    2384035.65182465 = beat(4635.99766796409,4627)
    2384110.34604552 = beat(74626.0277273697,72361.0252351259) = 360*60*60 / 0.543599
    La2011 Table 5

  2. Paul Vaughan says:

    moderators: calculations caught in filter

    _
    easy hindsight:
    https://tallbloke.wordpress.com/2013/01/09/tim-cullen-solar-system-holocene-lawler-events/

  3. Paul Vaughan says:

    algebraic review
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)
    49962.9015304794 = 1/(2/11.8627021700857-2/29.4701958106261-2/11.8619993833167+2/29.4571726091513)
    4669.65169718707 = 2/(-3/11.8627021700857+1/29.4701958106261+1/84.0331316671926+1/164.793624044745+4/11.8619993833167-4/29.4571726091513)

    simple alternate perspective
    anomalistic
    19.8549641949401 = 1/(1/11.8627021700857-1/29.4701958106261)
    13.8125825263028 = 1/(1/11.8627021700857-1/84.0331316671926)
    12.7828803855253 = 1/(1/11.8627021700857-1/164.793624044745)
    9.95061383963391 = 2/(3/11.8627021700857-1/29.4701958106261-1/84.0331316671926-1/164.793624044745)
    synodic
    9.92945505108639 = 2/(+4/11.8619993833167-4/29.4571726091513)
    4669.65169718707 = 2/(-3/11.8627021700857+1/29.4701958106261+1/84.0331316671926+1/164.793624044745+4/11.8619993833167-4/29.4571726091513)

    link to net search result for perihelion 4670

    _
    moderators: other calculations caught in filter

  4. Paul Vaughan says:

    4670 years

    This is a test comment.
    3 comments have vanished. Cause unknown

    headin’ up the mountain now.

    [mod] nothing in the WP spam filter now

  5. Paul Vaughan says:

    Mayan 36750

    15009.1608487337 = 5482096 / 365.25
    36135.2404360745 = beat(25672.5169367299,15009.1608487337)

    anomalistic
    171.471519050756 = beat(164.793624044745,84.0331316671926) — Standish

    tropical
    171.444289533663 = beat(163.7232045,83.74740682) — Seidelmann
    171.444286952825 = beat(163.723203285421,83.7474058863792) — ‘factsheet’

    1079630.33859387 = beat(171.471519050756,171.444289533663) ; * 2 = 2159260.67718775
    1079528.00372651 = beat(171.471519050756,171.444286952825) ; * 2 = 2159056.00745301

    36750.2561948093 = beat(2159260.67718775,36135.2404360745)
    36750.3154881725 = beat(2159056.00745301,36135.2404360745)

    compare
    Berger 1988 Table 4 (based on Berger 1978)
    2166101.14285714 = beat(75259,72732)
    36748.2810485504 = beat(2166101.14285714,36135.2404360745)

  6. Paul Vaughan says:

    supplementary
    lunisolar with general precession
    13374613.0030966 = beat(25771.4533429313,25721.8900031954)
    25672.5169367299 = axial(13374613.0030966,25721.8900031954)

  7. Paul Vaughan says:

    no mystery left here
    36135.2438440821 = axial(2159056.00745389,36750.3190131859)
    15009.1614366987 = axial(36135.2438440821,25672.5169367299) = 5482096.21475421 / 365.25
    bias hindsight “120k orbital solutions” tunes with round-off a typo f(UN)code’n’PRrhymesnot:
    30031.0042303539 = beat(36750.3190131859,16526.3120307908)
    15009.1624987455 = axial(36135.25,25672.5169367299) = 5482096.6026668 / 365.25
    weather influence campaign or entertainment: unknown

  8. Paul Vaughan says:

    Seidelmann short-model sidereal UJS bias clarification
    30031 = 59*509 (lowest primorial+1 that’s not prime)
    33052.4924754047 = harmean(36750,30031) ~= 33052.5
    36750.0186045651 = beat(30031,16526.25)
    36750.0196670027 — La(2004a,2010a)average

  9. Paul Vaughan says:

    systematic bias review
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    16.9122914926352 = harmean(29.4474984673838,11.8626151546089)
    21.1746788367349 = beat(84.016845922161,16.9122914926352)
    321.183589283115 = slip(21.1746788367349,19.8650360864628)
    1908.55545325512 = slip(321.183589283115,19.8650360864628)
    33052.6240615815 = slip(1908.55545325512,321.183589283115)

    compare
    15009.1608487337 = 5482096 / 365.25
    36135.2404360745 = beat(25672.5169367299,15009.1608487337)
    36750.3154881723 = beat(2159056.00745389,36135.2404360745)
    33052.6200736723 = harmean(36750.3154881723,30031)

    again: the bias in the short-duration models is systematic
    it could easily be corrected by experts before publication
    it isn’t

    why?
    unknown

  10. Paul Vaughan says:

    supplementary
    30031 = 13*11*7*5*3*2 + 1 = 13# + 1 = 59 * 509
    36750.3253483715 = beat( 30031 , 33052.6240615815 / 2)
    36135.2499689834 = axial(2159056.00745389,36750.3253483715) ~= 36135.25
    15009.1624933944 = axial(36135.2499689834,25672.5169367299) = 5482096.6007123 / 365.25
    recall:
    33053√Φ/8 ~= 5256; 5256√Φ/8 ~= 836

  11. Paul Vaughan says:

    typo: “/√Φ” not “√Φ”
    serpent no. anomalistic UN guidance

  12. Paul Vaughan says:

    Lunisolar Bias

    general & lunisolar precession
    25746.6478202264 = harmean(25771.4533429313,25721.8900031954)

    NASA ‘factsheet’ tropical
    11.8619854620833 = beat(25746.6478202264,11.8565229295003)
    29.4571820908507 = beat(25746.6478202264,29.4235181382615)

    19.858866774147 = beat(29.4571820908507,11.8619854620833)
    60.9467636123559 = slip(29.4571820908507,11.8619854620833)
    883.349939238609 = slip(60.9467636123559,19.858866774147) ; / 2 = 441.674969619304

    600.349139225674 = harmean(936.955612197409,441.674969619304) ; * 4 = 2401.3965569027
    835.54616509501 = beat(936.955612197409,441.674969619304)

    biased (short-duration) Seidelmann sidereal model:
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    61.0464822565173 = slip(29.4474984673838,11.8626151546089)
    835.546575435631 = slip(61.0464822565173,19.8650360864628)

    2401.00140862743 = harmean(2401.3965569027,2400.60639037357)
    2401.00140862743 = 7.00000102669615 ^ 4

    supplementary anomalistic (Standish 1992 Table 2a) review
    936.955612197409 = 1/(-2/11.8627021700857+5/29.4701958106261)
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)
    1536.74746987137 = harmean(4270.51884168654,936.955612197409)
    2400.60639037357 = beat(1536.74746987137,936.955612197409)

  13. Paul Vaughan says:

    Why aren’t the biased models unbiased before publication?
    Unknown.

    25770.0359146014 = 360*60*60/50.290966 — widely cited general precession rate
    25722.1631216381 = 360*60*60/50.38456501 — W94 lunisolar precession

    25746.0772642216 = harmean(25770.0359146014,25722.1631216381)

    13846300.2974074 = beat(25770.0359146014,25722.1631216381)
    25674.4678646892 = axial(13846300.2974074,25722.1631216381)

    36131.3759941719 = beat(25674.4678646892,15009.1608487337) — spicy whether
    15009.1608487337 = 5482096 / 365.25

    11.8619855831937 = beat(25746.0772642216,11.8565229295003)
    29.4571828377274 = beat(25746.0772642216,29.4235181382615)

    19.858866774147 = beat(29.4571828377274,11.8619855831937)
    60.9467604151707 = slip(29.4571828377274,11.8619855831937)
    883.351954142505 = slip(60.9467604151707,19.858866774147)
    835.54977055391 = beat(936.955612197409,441.675977071252)

    36130.8944646852 = beat(29.447498973306,29.4235181382615) — Sidorenkov
    36131.2752839187 = harmean(36131.65611118,36130.8944646852) —- or
    36131.65611118 = beat(29.4474984673838,29.4235181382615) — Seidelmann

    29.447499867163 = beat(36131.3759941719,29.42351935)
    11.862615400484 = 2 * beat( 835.54977055391 , 29.447499867163 / 5 )

    blur 64k mixmmod11sample
    23093.6961437637 = beat(11.862615400484,11.85652502)
    63999.6554631887 = beat(36131.3759941719,23093.6961437637) ~= 64k
    un weather seidelenkov or sidormann

    11.8626149080812 = 2 * beat( 835.54977055391 , 29.4474986534485 / 5 )
    29.4474986534485 = beat(36131.3759941719,29.4235181382615)

    Bound 2401 = 7^4 with surely enough error = 0.000014667% J-aims.

  14. Paul Vaughan says:

    2362 notes

    comparing systematically biased models
    (which can be systematically unbiased)

    Seidelmann (1992) & Sidorenkov (2017) share superficial similarity
    however, lots of detail differences are noteworthy
    here’s one to start with

    Seidelmann (1992) sidereal review
    =
    Jovian V-E 5256 Ratio Summary
    J:U:S:N = 1:2:16:31

    J: 1 = ⌊0.999878363433384⌉ = ⌊5256 / 5256.6393995685⌉
    S: 16 = ⌊16.0035845857963⌉ = ⌊5256 / 328.426420457382⌉
    U: 2 = ⌊1.99911615294986⌉ = ⌊5256 / 2629.16188848974⌉
    N: 31 = ⌊31.0045597971227⌉ = ⌊5256 / 169.52345185329⌉

    J: 5256.6393995685 = 1 * 5256.6393995685
    S: 5254.8227273181 = 16 * 328.426420457382
    U: 5258.32377697949 = 2 * 2629.16188848974
    N: 5255.227007452 = 31 * 169.52345185329
    =

    Sidorenkov analogs:
    5256 / x =
    1.11245625125401
    16.039288833368
    1.98119882107214
    31.0141327829003

    4724.68017872629 = 1 * 4724.68017872629 — carefully take note of something about this one
    5243.12523289982 = 16 * 327.695327056239
    5305.87838443763 = 2 * 2652.93919221881
    5253.60490137049 = 31 * 169.471125850661

  15. Paul Vaughan says:

    even more perfect jupiter-saturn framing — part i

    19.8650352019356 = beat(29.447498973306,11.8626149212868)
    9.9325176009678 = 19.8650352019356 / 2
    4.9662588004839 = 19.8650352019356 / 4

    16.9122913389518 = harmean(29.447498973306,11.8626149212868)
    8.4561456694759 = axial(29.447498973306,11.8626149212868)
    4.22807283473795 = 8.4561456694759 / 2
    2.11403641736897 = 8.4561456694759 / 4

    6.56961469713012 = axial(14.723749486653,11.8626149212868)

    10.7425999661684 = beat(16.9122913389518,6.56961469713012)
    5.3712999830842 = 10.7425999661684 / 2
    2.6856499915421 = 10.7425999661684 / 4

    4.73161069513687 = axial(16.9122913389518,6.56961469713012)
    2.36580534756843 = 4.73161069513687 / 2
    1.18290267378422 = 4.73161069513687 / 4

    131.716314078385 = slip(19.8650352019356,10.7425999661684)
    65.8581570391927 = slip(19.8650352019356,5.3712999830842)
    50.0715810605726 = slip(19.8650352019356,2.6856499915421)

    100.143162121145 = slip(19.8650352019356,4.73161069513687)
    50.0715810605727 = slip(19.8650352019356,2.36580534756843)
    96.1826326350372 = slip(19.8650352019356,1.18290267378422)

    supplementary
    61.0464717290582 = slip(29.447498973306,11.8626149212868)
    835.550927105133 = slip(61.0464717290582,19.8650352019356)

  16. Paul Vaughan says:

    even more perfect jupiter-saturn framing — part ii

    basic
    2432.00637869116 = slip(100.143162121145,19.8650352019356)
    1216.00318934558 = slip(100.143162121145,9.9325176009678)
    1216.00318934592 = slip(50.0715810605726,9.9325176009678)
    608.00159467279 = slip(100.143162121145,4.9662588004839)
    608.00159467296 = slip(50.0715810605726,4.9662588004839)
    608.001594672738 = slip(96.1826326350372,19.8650352019356)
    304.000797336369 = slip(96.1826326350372,9.9325176009678)
    compound
    1216.00318934576 = slip(356.529955086161,131.716314078385)
    1216.00318934575 = slip(208.887731776281,65.8581570391927)
    1216.00318934571 = slip(208.887731776281,50.0715810605726)
    1216.00318934591 = slip(104.44386588814,50.0715810605726)
    608.00159467288 = slip(178.26497754308,65.8581570391927)
    1216.00318934554 = slip(104.443865888141,50.0715810605726)

    a little more wholesome than “perfect jupiter-saturn framing”
    2432.00637869017 = 2/(27/11.8626149212868-67/29.447498973306)
    608.001594672542 = 1/(54/11.8626149212868-134/29.447498973306)
    19.0000498335169 = 1/(1728/11.8626149212868-4288/29.447498973306)

    19 ~= 1/(1728/j-4288/s)
    431 ~= 1/(-89/j+221/s)

    431.004429758615 = 1/(-89/11.8626149212868+221/29.447498973306)
    862.008859517229 = 2/(-89/11.8626149212868+221/29.447498973306)

    basic
    862.008859517104 = slip(131.716314078385,4.22807283473795)
    431.004429758552 = slip(131.716314078385,2.11403641736897)
    431.004429758552 = slip(65.8581570391927,2.11403641736897)
    compound
    862.008859517073 = slip(356.529955086161,65.8581570391927)
    862.008859517088 = slip(310.94560772299,131.716314078385)
    862.008859517088 = slip(155.472803861495,131.716314078385)
    431.004429758544 = slip(155.472803861495,65.8581570391927)

  17. Paul Vaughan says:

    Theorrery Skeptic

    Doubtful weather there’s a trustworthy political party (or orrery).

    JEV ingredients (Bollinger 1952 method)
    1.59868960469858 = beat(1.00001741273101,0.615197262149213)
    0.799344802349289 = 1.59868960469858 / 2
    0.399672401174645 = 1.59868960469858 / 4

    0.761766202327597 = harmean(1.00001741273101,0.615197262149213)
    0.380883101163799 = axial(1.00001741273101,0.615197262149213)
    0.190441550581899 = 0.380883101163799 / 2
    0.0952207752909496 = 0.380883101163799 / 4

    0.814040380789063 = beat(11.8626149212868,0.761766202327597)
    0.407020190394532 = 0.814040380789063 / 2
    0.203510095197266 = 0.814040380789063 / 4

    0.715800563194352 = axial(11.8626149212868,0.761766202327597)
    0.357900281597176 = 0.715800563194352 / 2
    0.178950140798588 = 0.715800563194352 / 4

    44.2785528962259 = slip(1.59868960469858,0.814040380789063)
    22.1392764481129 = slip(1.59868960469858,0.407020190394532)
    11.0696382240565 = slip(1.59868960469858,0.203510095197266)

    6.84872428662588 = slip(1.59868960469858,0.715800563194352)
    3.42436214331294 = slip(1.59868960469858,0.357900281597176)
    24.1185188101801 = slip(1.59868960469858,0.178950140798588)

    thus:

    350.939503542819 = slip(22.1392764481129,0.761766202327597)
    207.994354394449 = slip(44.2785528962259,0.399672401174645)
    207.994354394443 = slip(73.0136987128728,44.2785528962259)
    146.027397425746 = slip(44.2785528962259,1.59868960469858)
    73.0136987128728 = slip(22.1392764481129,0.799344802349289)

    89.3948929753992 = slip(44.2785528962259,0.190441550581899)
    88.5571057924501 = harmean(89.3948929753992,87.7348758857048) ; 44.2785528962251 = axial
    87.7348758857048 = slip(22.1392764481129,0.190441550581899)

    4724.68017871817 = beat(89.3948929753992,87.7348758857048)
    4724.68017872629 = slip(44.2785528962259,0.0952207752909496)
    4724.68017870897 = slip(350.939503542819,44.2785528962259)
    4724.68017872737 = slip(89.3948929753992,44.2785528962259)
    4724.68017870897 = slip(87.7348758857048,44.2785528962259)
    2362.34008935448 = slip(350.939503542819,22.1392764481129) ——— 700,44
    2362.34008936368 = slip(89.3948929753992,22.1392764481129) ——— notice
    2362.34008935448 = slip(87.7348758857048,22.1392764481129) ——— anything
    1181.17004467724 = slip(350.939503542819,11.0696382240565)
    1181.17004468184 = slip(89.3948929753992,11.0696382240565)
    1181.17004467724 = slip(87.7348758857048,11.0696382240565)

    980.292586153988 = slip(292.054794851491,88.5571057924518) ; 292 = 163+67+43+19
    490.146293076994 = slip(146.027397425746,44.2785528962259)
    245.073146538497 = slip(73.0136987128728,22.1392764481129)

  18. Paul Vaughan says:

    with NASA Horizons 1929.72222222222 sidereal
    JSUN orbital invariant (long-run wide-guassian central limit)
    2320.40158057186 = axial(130705.116382548,2362.3400894)
    compare (Horizons 1929.72222222222 JSUN orbital invariants only — without
    Sidorenkov JEV)
    2320.22208286955 = axial(130705.116382548,2362.15404490606)

    notin’ a few interresstin’ properties not shared by Seidelmann & Sidorenkov
    ˚Knot f(eel)in’PRsure’22DC(44)ide weather appearance orrery’a11 IT

  19. Paul Vaughan says:

    /a typo h/tM11
    130704.452624679 = beat(173901.37537739,74619.9907876555)
    73: lowest prime congruent to 1 mod 24
    73 = average(19,43,67,163) ; 19 = x mod 24 for x = 19, 43, 67, 163
    5256 = 7920 – 2400 – 240 – 24

  20. Paul Vaughan says:

    Perfect Contrusst?

  21. Paul Vaughan says:

    Simple Reason: Center Unknown

    Here’s the backstory on how the lunisolar bias stood out distinctly.
    One – and only one – of the many parameter lists fits the criterion.

    “[…] the largest known number not of the form a^2+s with s a semiprime”

    883.339228237648 = harmean( 936.716909730743 , 835.716909730743 )
    883.339228237648 = harmean ( 101 + C√φ , C√φ ) = 2 / ( 1/(101+C√φ) + 1/(C√φ) )

    101: lowest odd prime Mertens zero-crossing

    _____________
    supplementary

    1/(
    5/beat( harmean(general,lunisolar precession) , Saturn tropical) –
    2/beat( harmean(general,lunisolar precession) , Jupiter tropical)
    )

    low
    Seidelmann (1992) tropical
    883.332192747065 — LLR
    883.334207569974

    ~Center:
    883.342073334256 = grand harmonic mean

    high
    NASA ‘factsheet’ tropical
    883.349939238609 — LLR
    883.351954142505

  22. Paul Vaughan says:

    Diversifying Bidecadal Chandler Resonance Diagnostics

    Seidelmann (1992) short-duration (biased) sidereal
    6.46811773461345 = beat(1.18290267954072,1.00001743371442)

    Standish (1992) nodal
    19.8630730877524 = beat(29.4511026866654,11.862499899747)

    bias diagnostics
    280.076577904465 = slip(19.8630730877524,6.46811773461345)
    2790.5838772567 = slip(280.076577904465,19.8630730877524)

    Standish (1992) anomalistic orbital invariant (0=-2+5+3-6)
    2790.83682251396 = 1/(-2/11.8627021700857+5/29.4701958106261+3/84.0331316671926-6/164.793624044745)

    reverse-engineering sidereal earth estimate from jovian anomalistic:
    1.00001743139444 = 14/(2/11.8627021700857-5/29.4701958106261-3/84.0331316671926+6/164.793624044745+84/11.8626151546089+140/29.4474984673838+43/11.862499899747-43/29.4511026866654)

    Experts failed us severely more than a decade ago.
    They harassed us viciously and relentlessly without ever volunteering the most important information.
    Why? Unknown.

    compare – using Standish (1992) short-duration (biased) sidereal earth
    6.46811767349981 = beat(1.18290267954072,1.0000174322536)
    280.076463317364 = slip(19.8630730877524,6.46811767349981)
    2790.74314407649 = slip(280.076463317364,19.8630730877524)

  23. Paul Vaughan says:

    Anomalistic QBO Halstatt

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) —– Standish (1992) anomalistic

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    8.45806059760692 = axial(29.4701958106261,11.8627021700857)

    6.57189983390616 = axial(14.735097905313,11.8627021700857)

    9.46619163320184 = harmean(16.9161211952138,6.57189983390616)
    4.73309581660092 = axial(16.9161211952138,6.57189983390616)
    2.36654790830046 = 4.73309581660092 / 2
    1.18327395415023 = 4.73309581660092 / 4

    203.72293652095 = slip(19.8549641949401,9.46619163320184)
    50.9307341302375 = slip(19.8549641949401,2.36654790830046) ——————–

    4724.17556802657 = slip(203.72293652095,16.9161211952138)
    2362.08778401328 = slip(203.72293652095,8.45806059760692)

    4724.17556802739 = slip(50.9307341302375,16.9161211952138)
    2362.0877840137 = slip(50.9307341302375,8.45806059760692) ——————– 2362

    _
    2320.15794438421 = axial(130704.452624679,2362.0877840137)
    2320.15815353613 = axial(130705.116382548,2362.0877840137)

  24. Paul Vaughan says:

    anomalistic JEV intro

    0.761769224080824 = harmean(1.0000262476142,0.615197860179071)
    0.715803548953639 = axial(11.8627021700857,0.761769224080824)
    6.84967828238651 = slip(1.59867106414771,0.715803548953639)
    835.563824740778 = slip(6.84967828238651,0.761769224080824)

    1.59867106414771 = beat(1.0000262476142,0.615197860179071)
    24.0670906604158 = slip(6.84967828238651,1.59867106414771)
    24.067904774739 = ⌊(e^√7π)^(1/p)⌉^p – e^√7π for p=2,3,4,6,12

  25. Paul Vaughan says:

    Anomalistic “4670 years”

    19.8549641949401 = beat(29.4701958106261,11.8627021700857)

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)
    19.9477072615632 = beat(111.307357343015,16.9161211952138) ; / 2 = 9.97385363078158
    4270.51884168622 = slip(19.9477072615632,19.8549641949401)

    8.45806059760692 = axial(29.4701958106261,11.8627021700857)
    55.6536786715076 = axial(164.793624044745,84.0331316671926)
    9.97385363078158 = beat(55.6536786715076,8.45806059760692)
    2135.25942084311 = slip(19.8549641949401,9.97385363078158)

    4670.79911370059 = slip(2135.25942084311,19.8549641949401)

    invariant 0=1-3+1+1
    2135.25942084327 = 1/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)

  26. Paul Vaughan says:

    What’s in anomalistic “mode 11”? by US Diss Cover:

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857) ; / 2 = 9.92748209747005
    60.8544553085225 = slip(29.4701958106261,11.8627021700857)
    thus base what follows with Standish (1992) anomalistic

    _

    compare & contrast

    —-

    1.

    Seidelmann (1992) short-duration sidereal
    11.8626151546089
    29.4474984673838
    84.016845922161
    164.791315640078
    29.3625733662893 = harmonic mean

    19.8060547427555 = axial(60.8544553085225,29.3625733662892)
    66.1523612443866 = slip(28.3699051701599,19.8549641949401)
    8040.33760745923 = slip(19.8549641949401,19.8060547427555)

    172978.748617708 = slip(8040.33760745923,19.8549641949401) —– further notes be low
    86489.3743088542 = slip(8040.33760745923,9.92748209747005)

    56.7398103403199 = beat(60.8544553085225,29.3625733662892) ; / 2 = 28.3699051701599
    1182607.44767058 = slip(172978.748617708,66.1523612443866) ; * 2 = 2365214.89534115

    202615.037187354 = slip(86489.3743088542,66.1523612443866) ; * 2 = 405230.074374708

    —-

    2.

    Sidorenkov (2017)
    11.8626149212868
    29.447498973306
    84.0168377823409
    164.791321013005
    29.3625729287642 = harmonic mean

    analogUS (to 1. above) calculations rightly left as an exercise in undersstandin’ 4 those curryUS’n’willin’ (who’ll thus ‘no. who’ tune$ what – CO[$] ITsnot what y/n0boughtWHOyen owe…)

    —-

    supplementary

    2364963.50364963 = beat(74619.9907876555,72337.575351641)
    1182481.75182481 = beat(73001.7461837436,68756.9632341238)

    405629.613215262 = beat(304406.35241565,173901.37537739)

    172826.54615749 = beat(68756.9632341238,49188.0779029847)
    173901.37537739 = 1 / g_2

    in agree mint with us?
    Weather 5256 & 4724 are (no. doubt?) in agree mean T.

  27. Paul Vaughan says:

    JPLat0˚Know? Anomalistic weather variant. Mnemonic in-put taxicab:

    Seidelmann (1992) tropical
    4.72860952101702 = 2/(3/11.85652502+5/29.42351935) = 1727.12462755147 d ~= 1727
    2.36430476050851 = 1/(3/11.85652502+5/29.42351935) = 863.562313775734 d
    1.18215238025426 = 1/(6/11.85652502+10/29.42351935) = 431.781156887867 d
    0.591076190127128 = 1/(12/11.85652502+20/29.42351935) = 215.890578443933 d

    “Euler started to use the letter e for the constant in 1727 or 1728 […]”

    Sidorenkov (2017)
    4.73161069513687 = 2/(3/11.8626149212868+5/29.447498973306) = 1728.22080639874 d
    2.36580534756843 = 1/(3/11.8626149212868+5/29.447498973306) = 864.110403199371 d
    1.18290267378422 = 1/(6/11.8626149212868+10/29.447498973306) = 432.055201599685 d
    0.591451336892109 = 1/(12/11.8626149212868+20/29.447498973306) = 216.027600799843 d

    Seidelmann (1992) sidereal short-duration model
    4.73161071816289 = 2/(3/11.8626151546089+5/29.4474984673838) = 1728.220814809 d
    2.36580535908144 = 1/(3/11.8626151546089+5/29.4474984673838) = 864.110407404498 d
    1.18290267954072 = 1/(6/11.8626151546089+10/29.4474984673838) = 432.055203702249 d
    0.591451339770361 = 1/(12/11.8626151546089+20/29.4474984673838) = 216.027601851124 d

    Seidelmann (1992) synodic
    4.73208801967701 = 2/(3/11.8619993833167+5/29.4571726091513) = 1728.39514918703 d
    2.36604400983851 = 1/(3/11.8619993833167+5/29.4571726091513) = 864.197574593514 d
    1.18302200491925 = 1/(6/11.8619993833167+10/29.4571726091513) = 432.098787296757 d
    0.591511002459627 = 1/(12/11.8619993833167+20/29.4571726091513) = 216.049393648379 d

    Horizons 1929.72222222222 sidereal (wide-Guassian sample-center)
    4.73208337117502 = 2/(3/11.8619848807702+5/29.4571542179636) = 1728.39345132168 d
    2.36604168558751 = 1/(3/11.8619848807702+5/29.4571542179636) = 864.196725660838 d
    1.18302084279375 = 1/(6/11.8619848807702+10/29.4571542179636) = 432.098362830419 d
    0.591510421396877 = 1/(12/11.8619848807702+20/29.4571542179636) = 216.049181415209 d

    Standish (1992) nodal
    4.73181582715489 = 2/(3/11.862499899747+5/29.4511026866654) = 1728.29573086832 d
    2.36590791357745 = 1/(3/11.862499899747+5/29.4511026866654) = 864.147865434162 d
    1.18295395678872 = 1/(6/11.862499899747+10/29.4511026866654) = 432.073932717081 d
    0.591476978394361 = 1/(12/11.862499899747+20/29.4511026866654) = 216.03696635854 d

    Standish (1992) anomalistic
    4.73309581660092 = 2/(3/11.8627021700857+5/29.4701958106261) = 1728.76324701349 d ~= 1729
    2.36654790830046 = 1/(3/11.8627021700857+5/29.4701958106261) = 864.381623506743 d
    1.18327395415023 = 1/(6/11.8627021700857+10/29.4701958106261) = 432.190811753372 d
    0.591636977075115 = 1/(12/11.8627021700857+20/29.4701958106261) = 216.095405876686 d

  28. Paul Vaughan says:

    Solves Longstanding Curiosity

    This is highly technical but precise.

    26256.8112288057 = slip(8040.33760745923,16.9161211952138)
    306006.586378586 = slip(26256.8112288057,66.1523612443866)

    The original curiosity? Systematically be low.

    397906.462823915 = beat(306006.586378586,172978.748617708) ; / 4 =
    99476.6157059787
    99476.8155050052 = beat(16.9161211952138,16.9132450828034)
    16.9132450828034 = harmean(29.4571309198874,11.861990807677) —- Standish (1992) sidereal

    (further) clarifies lunisolar bias; well-paid technicians “No!” managers can unbias (hierarchically nested) models for clean, simple public presentation
    _
    supplementary
    398.773134809729 = slip(56.7398103403199,19.8549641949401)
    935.113131399583 = slip(398.773134809729,16.9161211952138)
    936.033465118244 = harmean(936.955612197393,935.113131399583) ~= 936.0
    for future reference
    2365.09305036237 = slip(398.773134809729,9.92748209747005)
    4730.18610072473 = slip(398.773134809729,19.8549641949401)

  29. Paul Vaughan says:

    Adjust Sidorenkov (2017) earth sidereal year estimate (which looks like a long-duration estimate) to 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) to match SUNEV 5256 (but not JEV 5256) more closely than Seidelmann (1992).

    At this point a very large volume of calculations needs to be presented to crystallize the lunisolar bias in detail. Presently this isn’t feasible.

  30. Paul Vaughan says:

    To complete the transformation:

    11.862615918328 = 2/(1/((φ^22+1/11)^(e/11+1/22))+5/((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))-3/0.615197262149213)

    0.761766208470514 = harmean(1.00001743390371,0.615197262149213)
    0.814040383108898 = beat(11.862615918328,0.761766208470514)
    1.59868955058708 = beat(1.00001743390371,0.615197262149213)

    44.2784630136991 = slip(1.59868955058708,0.814040383108898)
    351.268593378986 = slip(44.2784630136991,0.761766208470514)
    5256.18499467858 = slip(351.268593378986,44.2784630136991)

    alternately:
    0.380883104235257 = axial(1.00001743390371,0.615197262149213) ; / 4 = 0.0952207760588143
    5256.18499465625 = slip(44.2784630136991,0.0952207760588143)

  31. Paul Vaughan says:

    100ka note

    19.8549641949401 = beat(29.4701958106261,11.8627021700857) —- Standish (1992) anomalistic
    19.8589101021728 = beat(29.4571726091513,11.8619993833167) —– Seidelmann (1992) synodic
    99925.8030607636 = beat(19.8589101021728,19.8549641949401)

    99972.391587704 = harmean(100019.023576957,99925.8030607636) ———— v ————–

    100019.023576957 = beat(16.9161211952138,16.9132606717144)
    16.9132606717144 = harmean(29.4571726091513,11.8619993833167) —- Seidelmann (1992) synodic
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857) — Standish (1992) anomalistic
    8.45806059760692 = axial(29.4701958106261,11.8627021700857) —– Standish (1992) anomalistic
    _

    55.6536786715076 = axial(164.793624044745,84.0331316671926) —– Standish (1992) anomalistic
    111.307357343015 = harmean(164.793624044745,84.0331316671926) — Standish (1992) anomalistic
    171.471519050756 = beat(164.793624044745,84.0331316671926)

    19.9477072615632 = beat(111.307357343015,16.9161211952138) —– Bollinger (1952) method
    9.97385363078158 = beat(55.6536786715076,8.45806059760692)

    4270.51884168622 = slip(19.9477072615632,19.8549641949401) —- orbital invariant
    2135.25942084311 = slip(19.8549641949401,9.97385363078158) —- orbital invariant

    49831.9228185121 = slip(4270.51884168622,19.8549641949401)
    4670.79911370059 = slip(2135.25942084311,19.8549641949401) —- “4670 years” (review)

    45051.2652209889 = slip(4270.51884168622,8.45806059760692)
    99973.0172242633 = slip(45051.2652209889,4270.51884168622) —————- ^ ————-

  32. Paul Vaughan says:

    Selected Algebra

    Seidelmann (1992) synodic:
    11.8619993833167, 29.4571726091513
    Standish (1992) anomalistic:
    11.8627021700857, 29.4701958106261, 84.0331316671926, 164.793624044745
    _

    “4670 years”

    4669.65169718707 = 2/(-3/11.8627021700857+1/29.4701958106261+1/84.0331316671926+1/164.793624044745+4/11.8619993833167-4/29.4571726091513)

    4670.79911381622 = 1/(107/11.8627021700857-323/29.4701958106261+108/84.0331316671926+108/164.793624044745)
    _

    “100 kiloyears”

    99972.3915878471 = 4/(1/11.8627021700857-3/29.4701958106261-1/11.8619993833167+3/29.4571726091513)

    99973.0183503139 = 1/(2766/11.8627021700857-8342/29.4701958106261+2777/84.0331316671926+2777/164.793624044745)
    _

    “41 kiloyears”

    derivation
    8.45806059760692 = axial(29.4701958106261,11.8627021700857) ; / 4 =
    2.11451514940173
    45051.2652209889 = slip(4270.51884168622,8.45806059760692) ; / 4 =
    11262.8163052472 = slip(4270.51884168622,2.11451514940173)
    41002.979235297 = slip(11262.8163052472,2135.25942084311)

    summary:
    41002.9789528914 = 1/(-5029/11.8627021700857+15167/29.4701958106261-5049/84.0331316671926-5049/164.793624044745)
    _

    4724 years &
    2362 years

    derivations

    14.6844304038441 = axial(111.307357343015,16.9161211952138) ; / 4 =
    3.67110760096102
    48.6117359109827 = slip(19.8549641949401,3.67110760096102)
    384.86912152873 = slip(48.6117359109827,16.9161211952138)
    4724.17556801612 = slip(2135.25942084311,384.86912152873)

    19.9477072615632 = beat(111.307357343015,16.9161211952138) ; / 4 =
    4.98692681539079
    1067.62971042156 = slip(19.8549641949401,4.98692681539079)
    4724.17556801612 = slip(1067.62971042156,384.86912152873)

    192.434560764365 = slip(48.6117359109827,8.45806059760692)
    2362.08778400806 = slip(1067.62971042156,192.434560764365)

    algebra:
    2362.08778401334 = 1/(31/11.8627021700857-77/29.4701958106261)
    4724.17556802668 = 2/(31/11.8627021700857-77/29.4701958106261)

  33. Paul Vaughan says:

    tropical
    4724 years &
    2362 years (orbital invariant: 0=-2+5+3-6)

    Seidelmann (1992) tropical
    2361.92512664087 = 1/(-2/11.85652502+5/29.42351935+3/83.74740682-6/163.7232045)
    4723.85025328173 = 2/(-2/11.85652502+5/29.42351935+3/83.74740682-6/163.7232045)

    NASA ‘factsheet’ tropical
    2362.05130068208 = 1/(-2/11.8565229295003+5/29.4235181382615+3/83.7474058863792-6/163.723203285421)
    4724.10260136417 = 2/(-2/11.8565229295003+5/29.4235181382615+3/83.7474058863792-6/163.723203285421)

  34. Paul Vaughan says:

    adjusting Seidelmann (1992) short-duration sidereal earth to a long-duration 4724 year tuning:
    4723.99999667307 = 1/(-693.5/0.615197263396975+1166.5/1.00001741532595-465/11.8626151546089)
    1.00001741532595 = 365.256360947803 days ; compare:
    1.0000174152119 = 365.256360906146 days —- Standish (1992) sidereal

  35. Paul Vaughan says:

    easy hindsight

    supplementary notes on geophysical 64 year structure previously illustrated & explored
    Standish nodal with day
    3.93839482956483 = slip(1.00001071395229,0.0027378507871321)
    63.8858575898369 = slip(3.93839482956483,1.00001071395229) ~= 64 year
    Meeus & Savoie tropical with day
    4.12891838459878 = slip(0.999978614647502,0.0027378507871321)
    32.0054607170862 = slip(4.12891838459878,0.999978614647502) ~= 32 year

    supplementary note on 26 (& 52) year geophysical structure(s) previously noted
    Standish anomalistic with day
    3.85237502099482 = slip(1.0000262476142,0.0027378507871321)
    26.077823955957 = slip(3.85237502099482,1.0000262476142) ~= 26 year

  36. Paul Vaughan says:

    JSEV tropical (Seidelmann 1992)

    1.59868953279706 = beat(0.99997862,0.61518257)
    0.799344766398529 = 1.59868953279706 / 2
    0.399672383199264 = 1.59868953279706 / 4

    0.761743683794994 = harmean(0.99997862,0.61518257)
    0.380871841897497 = axial(0.99997862,0.61518257)
    0.190435920948749 = 0.380871841897497 / 2
    0.0952179604743743 = 0.380871841897497 / 4

    61.0914225103732 = slip(29.42351935,11.85652502)

    0.771361726706669 = beat(61.0914225103732,0.761743683794994)
    0.385680863353334 = 0.771361726706669 / 2
    0.192840431676667 = 0.771361726706669 / 4

    1.50472508020829 = harmean(61.0914225103732,0.761743683794994)
    0.752362540104143 = axial(61.0914225103732,0.761743683794994)
    0.376181270052072 = 0.752362540104143 / 2
    0.188090635026036 = 0.752362540104143 / 4

    12.800522799798 = slip(1.59868953279706,0.752362540104143)
    6.40026139989901 = slip(1.59868953279706,0.376181270052072)
    3.2001306999495 = slip(1.59868953279706,0.188090635026036)

    1859.26427454788 = slip(12.800522799798,1.59868953279706)
    1859.26427454788 = slip(6.40026139989901,1.59868953279706)
    1859.26427454788 = slip(3.2001306999495,1.59868953279706)

    929.632137273939 = slip(12.800522799798,0.799344766398529)
    929.632137273939 = slip(6.40026139989901,0.799344766398529)
    929.632137273939 = slip(3.2001306999495,0.799344766398529)

    59.0007181299284 = slip(12.800522799798,0.190435920948749)
    29.5003590649642 = slip(12.800522799798,0.0952179604743743)
    29.5003590649642 = slip(6.40026139989901,0.0952179604743743)

  37. Paul Vaughan says:

    JSUN tropical (Seidelmann 1992) 1470 year & 2402 year

    8.45107360405992 = axial(29.42351935,11.85652502)
    16.9021472081198 = harmean(29.42351935,11.85652502)
    19.8588720868409 = beat(29.42351935,11.85652502)

    3635.42278750964 = slip(163.7232045,83.74740682)
    17760.7407596846 = slip(3635.42278750964,171.444289533663)

    16.9182475901445 = beat(17760.7407596846,16.9021472081198)
    8.45912379507225 = 16.9182475901445 / 2
    4.22956189753613 = 16.9182475901445 / 4

    33.7721548821359 = harmean(17760.7407596846,16.9021472081198)
    16.886077441068 = axial(17760.7407596846,16.9021472081198)
    8.44303872053399 = 16.886077441068 / 2
    4.22151936026699 = 16.886077441068 / 4

    114.253729166588 = slip(19.8588720868409,16.9182475901445)
    57.1268645832942 = slip(19.8588720868409,8.45912379507225)
    67.1361846326796 = slip(19.8588720868409,4.22151936026699)

    475.496204649037 = slip(114.253729166588,16.9021472081198)
    237.748102324519 = slip(114.253729166588,8.45107360405992)
    118.874051162259 = slip(57.1268645832942,4.22553680202996)

    2939.57946202776 = slip(114.253729166588,4.22553680202996) ~= 2940
    2939.57946202744 = slip(475.496204649037,114.253729166588)
    2939.57946202744 = slip(237.748102324519,114.253729166588)
    2939.57946202744 = slip(118.874051162259,114.253729166588)

    1469.78973101388 = slip(114.253729166588,2.11276840101498) ~= 1470
    1469.78973101372 = slip(475.496204649037,57.1268645832942)
    1469.78973101372 = slip(237.748102324519,57.1268645832942)
    1469.78973101372 = slip(118.874051162259,57.1268645832942)

    2402.06517243955 = slip(67.1361846326796,16.9021472081198) ~= 2402

  38. Paul Vaughan says:

    Amicable Bonds (tropical version)

    “consensus” of Seidelmann (1992) & NASA ‘factsheet’
    29.4235187441307 = harmean(29.42351935,29.4235181382615)
    11.8565239747501 = harmean(11.85652502,11.8565229295003)
    0.999978358596783 = harmean(0.99997862,0.999978097193703)
    0.615181976306751 = harmean(0.61518257,0.615181382614647)

    explore what others might have in mind:
    0.615181976285804 — tuned to “4670 years” (remember Bond pub. with 1800 & 4670 graph)
    0.999978614647502 — Meeus & Savoie (1992) rounded-off value used by so many

    1791.85469669027 = 1/(-8/11.8565239747501+16/29.4235187441307+13/0.615181976285804-21/0.999978614647502)

    1799.99949076613 = 221/220/(-8/11.8565239747501+16/29.4235187441307+13/0.615181976285804-21/0.999978614647502)

    1791.85469669027 = axial(395999.88796855,1799.99949076613)
    1791.85469669027 = axial( 220 * 1799.99949076613 , 1799.99949076613 )

    gives:

    1.59868553691508 = beat(0.999978614647502,0.615181976285804)
    0.799342768457541 = 1.59868553691508 / 2
    0.399671384228771 = 1.59868553691508 / 4

    0.761743227089166 = harmean(0.999978614647502,0.615181976285804)
    0.380871613544583 = axial(0.999978614647502,0.615181976285804)
    0.190435806772291 = 0.380871613544583 / 2
    0.0952179033861457 = 0.380871613544583 / 4

    0.771361261986248 = beat(61.0913999839243,0.761743227089166)
    0.385680630993124 = 0.771361261986248 / 2
    0.192840315496562 = 0.771361261986248 / 4

    1.50472418232299 = harmean(61.0913999839243,0.761743227089166)
    0.752362091161495 = axial(61.0913999839243,0.761743227089166)
    0.376181045580747 = 0.752362091161495 / 2
    0.188090522790374 = 0.752362091161495 / 4

    22.0353413519587 = slip(1.59868553691508,0.771361261986248)
    11.0176706759793 = slip(1.59868553691508,0.385680630993124)

    12.8009052113462 = slip(1.59868553691508,0.752362091161495)

    304.013638441169 = slip(22.0353413519587,0.761743227089166)
    152.006819220584 = slip(22.0353413519587,0.380871613544583)
    76.0034096102922 = slip(22.0353413519587,0.190435806772291)
    76 = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24 = 19 + 19 + 19 + 19

    1791.85469667879 = slip(12.8009052113462,1.59868553691508)
    1799.99949075461 = 221 / 220 * 1791.85469667879

    164.876143595902 = slip(22.0353413519587,0.399671384228771) ———– z
    4670.79912511218 = slip(164.876143595902,11.0176706759793)

    perfect: s(496) = 496 = s(652) = s(s(608))

    608.027276881365 = 2/(-29/11.8565239747501+58/29.4235187441307-44/0.615181976285804+72/0.999978614647502)

    1216.05455376273 = 4/(-29/11.8565239747501+58/29.4235187441307-44/0.615181976285804+72/0.999978614647502)

    2432.10910752546 = 8/(-29/11.8565239747501+58/29.4235187441307-44/0.615181976285804+72/0.999978614647502)

  39. Paul Vaughan says:

    1800 & 4670 years:
    weather myth or math?

    lunisolar precession with tropical “Consenzus”
    11.8619917685154 = beat(25721.8900031954,11.8565239747501) — LLR
    11.8619917394734 = beat(25722.0265616918,11.8565239747501) — harmean(LLR,W94)
    11.8619917104315 = beat(25722.1631216381,11.8565239747501) — W94

    with anomalistic heavyweight Jupiter
    198078.497253751 = beat(11.8627021700857,11.8619917685154) ; * 2 = 396156.994507502
    198070.399443988 = beat(11.8627021700857,11.8619917394734) ; * 2 = 396140.798887976
    198062.302295769 = beat(11.8627021700857,11.8619917104315) ; * 2 = 396124.604591539

    1799.99857738802 = beat(396156.994507502,1791.85700697991)
    1799.9989117563 = beat(396140.798887976,1791.85700697991)
    1799.99924612472 = beat(396124.604591539,1791.85700697991)
    gives 4672.37005613987 (using 0.615181976306751 = harmean(0.61518257,0.615181382614647))

    1799.9962460564 = beat(396156.994507502,1791.85469669027)
    1799.99658042381 = beat(396140.798887976,1791.85469669027)
    1799.99691479137 = beat(396124.604591539,1791.85469669027)
    gives 4670.7991 (using 0.615181976285804)

    a reverse view bringz estimate ll’un ‘uz ole lore PR’ cz!sun from tropical!Jupiter “consensus”

    with 0.615181976306751 = harmean(0.61518257,0.615181382614647) & 4672.37005613987 :
    396088.097411778 = beat(1800,1791.85700697991) ; / 2 = 198044.048705889
    11.8619916449528 = axial(198044.048705889,11.8627021700857)
    25722.4710182008 = beat(11.8619916449528,11.8565239747501)

    with 0.615181976285804 & 4670.7991 :
    395975.242590358 = beat(1800,1791.85469669027) ; / 2 = 197987.621295179
    11.8619914424619 = axial(197987.621295179,11.8627021700857)
    25723.4232260154 = beat(11.8619914424619,11.8565239747501)

    also note
    Standish sidereal with Standish anomalistic
    197810.936263778 = beat(11.8627021700857,11.861990807677) ; / 2 = 395621.872527555
    Standish sidereal with NASA ‘factsheet’ tropical
    25721.4885660692 = beat(11.861990807677,11.8565229295003)
    25721.8900031954 = LLR lunisolar precession
    11.8565229295003 = NASA ‘factsheet’ tropical Jupiter
    11.8565230147972 = axial(25721.8900031954,11.861990807677)

    4 those who (anomalistically) are averse (“Can˚T get from the Cab to the C[I]RB” — The ‘PR 10’ drs)
    2 amicable bonds

  40. Paul Vaughan says:

    Enough $aid Sir Pentagon

    terre$ketchof whether
    loony so lure

    buy USamicably
    COMBINE Sidorenkov (2017) short-duration sidereal
    WITH Standish long-duration anomalistic
    883.352600477237 = harmean(936.955612197409,835.550927105136)
    984.598022541318 = beat( 25771.4533429313 / 3 , 883.352600477237 )
    ˚T?hird harmonic of General precession arises from opposing trigon COefficients (5 – 2 = 3) in sidereal-tropical conversion (use to detect bias inside pub.lists)
    979.992217251443 = beat(984.598022541318,491.144860474028) ; * 75 / 2 =
    36749.7081469291 ; compare:
    36749.7014379182 — La2011 Table 6 La2010a

    lunisolar bias estimate using Seidelmann (1992) short-duration sidereal JSU
    491.144860474028 = 1/(-3187.5/11.8626151546089+10462.5/29.4474984673838-7275/84.016845922161-37.5/30031-3/25771.4533429313+1/883.339228237648)

    noteworthy: 50482100 (rounded-off version of 50482096)
    systematically links NA!SA ‘factsheet’ & Seidelmann (1992) tropical (no. mystery in such hindsite)

    0.999978614647502 widely-cited rounded-off Meeus & Savoie (1992) tropical
    0.99997861640616 = 31/(-3187.5/11.8626151546089+10462.5/29.4474984673838-7275/84.016845922161-37.5/30031+21/0.0745030006844627+10/0.0754402464065708-31/0.0808503463381246-3/25770.7446092762+1/883.339228237648)
    off by 1 minute (time no. T˚angle) per century
    0.999978616353183 unrounded Meeus & Savoie (1992)

    supplementary

    0 = -42.5 + 139.5 – 97
    33052.6240611784 = 1/(-42.5/11.8626151546089+139.5/29.4474984673838-97/84.016845922161)
    36750.3253473747 = 1/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031)
    36135.2499680199 = 1/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031+0.5/83.7474058863792-0.5/163.723203285421-0.5/84.0331316671926+0.5/164.793624044745)
    15009.1624932282 = 1/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031+0.5/83.7474058863792-0.5/163.723203285421-0.5/84.0331316671926+0.5/164.793624044745+2/25721.8900031954-1/25771.4533429313)
    5482096.60065159 = 365.25/(-85/11.8626151546089+279/29.4474984673838-194/84.016845922161-1/30031+0.5/83.7474058863792-0.5/163.723203285421-0.5/84.0331316671926+0.5/164.793624044745+2/25721.8900031954-1/25771.4533429313)

    0 = 31 – 21 – 10
    491.132481368366 = 1/(31*(1/0.999978614647502+1/0.0808503463381246)-21/0.0745030006844627-10/0.0754402464065708)
    491.145236548486 = 1/(31*(1/0.999978616353183+1/0.0808503463381246)-21/0.0745030006844627-10/0.0754402464065708)

    derive
    0.0748024157879311 = axial(0.999978616353183,0.0808503463381246)
    5.99685290323073 = beat(0.0754402464065708,0.0745030006844627)

    18.6129703214384 = beat(0.0748024157879311,0.0745030006844627)
    8.84735306511776 = beat(0.0754402464065708,0.0748024157879311)

    16.8627866218841 = beat(18.6129703214384,8.84735306511776)
    179.333487684639 = slip(18.6129703214384,8.84735306511776)
    ace shown
    491.145236537217 = slip(179.333487684639,16.8627866218841)

    984.581409347557 = beat( 25771.4533429313 / 3 , 883.339228237648 )
    984.584512785494 = beat( 25770.7446092762 / 3 , 883.339228237648 )
    984.587616242996 = beat( 25770.0359146014 / 3 , 883.339228237648 )

    980.01017321043 = beat(984.581409347557,491.145236537217)
    980.007098542353 = beat(984.584512785494,491.145236537217)
    980.004023893569 = beat(984.587616242996,491.145236537217)

    36750.3814953911 = 75 / 2 * 980.01017321043
    36750.2661953382 = 75 / 2 * 980.007098542353
    36750.1508960089 = 75 / 2 * 980.004023893569

    36135.3042523091 = axial(2159056.00745389,36750.3814953911)
    15009.1718585901 = axial(36135.3042523091,25672.5169367299)
    5482100.02135004

    review
    Berger 1988 Table 4 (based on Berger 1978)
    2166101.14285714 = beat(75259,72732)
    36748.2810485504 = beat(2166101.14285714,36135.2404360745)
    note$ trees in llUNe^don foresst matrix calllculaceyen
    2159056.00745301 = 2 * beat(171.471519050756,171.444286952825)
    SSTand!sh anomalistic with ‘factsheet’ tropical
    PC in
    fact ch.UK IT
    luke dawn sank$yuan tyrant$
    R out of ˚T ou ch! within UKquality

    36750.3190131859 = 1/(g_3+g_4) — La2021 Table 2 with explore a tory add “just” (weather mint or) mean˚T
    36135.2438440821 = axial(2159056.00745389,36750.3190131859)
    15009.1614366987 = axial(36135.2438440821,25672.5169367299)
    5482096.21475421 —- pub.list.cz!is note$baseofllreflect$yen=peace$ together quickly

  41. Paul Vaughan says:

    120,000 “noteworthy” Typo Solutions

    won over UN √8
    1800 ~= 1 / √8 * 5090
    4670 = s(4370) + (378-178)*2

    owe bowl’n’Valley
    323 = 196883-196560

    “bye don!” Jail˚Talk
    25746 = Σs(5090) – ΣΦ(323)
    25746 = Σs(5090) – ΣΦ(936)

    0 LA˚Table
    25722 = Σs(5090) – (378-178)*2 – 71
    25746 = Σs(5090) – (378-178)*2 – 47
    25770 = Σs(5090) – (378-178)/2 – 196883 + 196560

    too Sell a con
    25808 = Σs(5090) – 11#/3#
    25808 = Σs(5090) – 7*(28^2-27^2)
    25808 = Σs(5090) – 7*(1^2+2^2+3^2+4^2+5^2)

    ——————–

    s(5090) = 4090
    s(4090) = 3290
    s(3290) = 3622
    s(3622) = 1814
    s(1814) = 910
    s(910) = 1106
    s(1106) = 814
    s(814) = 554
    s(554) = 280
    s(280) = 440
    s(440) = 640
    s(640) = 890
    s(890) = 730
    s(730) = 602
    s(602) = 454
    s(454) = 230
    s(230) = 202
    s(202) = 104
    s(104) = 106
    s(106) = 56
    s(56) = 64
    s(64) = 63
    s(63) = 41
    s(41) = 1
    s(1) = 0

    th borg herd sum??un
    26193 = Σs(5090)
    ABout what nos.˚TemperUN

  42. Paul Vaughan says:

    317 years
    anomalistic JSU
    the very top-level cycle is a familiar one
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    21.1796477480038 = beat(84.0331316671926,16.9161211952138)
    317.450267066043 = slip(21.1796477480038,19.8549641949401)

  43. Paul Vaughan says:

    ok

    2545 = ΣΣδ(220)
    5090 = ΣΣδ(220) * 2

    unI˚Que.T˚A CO$metallica
    400 = s(2401) = 2*(378-178) = s(836-42) = s(496+28) = 744-104-240
    400 = ΣΦ(323) – Σφ(323) ; 447 = ΣΦ(323) ; 47 = Σφ(323)
    400 = Σs(242) ; 242 = ΣΣΔ(178)-ΣΣφ(178) = average(ΣΔ(220),Σδ(220)) = ΣΦ(163)-Σφ(163)
    993 = (σ^2)(400) ; Σδ(Σδ(993)) = Σδ(Σδ(Σδ(378))) = 902 = Σδ(894) ; Σδ(902) = 894
    σ(894) = σ(1691) = 1800 ; ΣΦ(1800) = 735 ; 600 = σ(216)
    216 = 378 – (163+67+43+19+28) + (10+13+18+22+37+58)

    J[amai$SUNami]11buy
    100 = average(-ΣΦ(220),Σφ(220)) = average(-178,378) = 71-Σδ(42)
    100 = average(-Σφ(323),ΣΦ(323))/2 = 4370-s(4370) = s(194)
    100 = 2+3+5+7+11+13+17+19+23 = Σ(primes up to 23)
    100 = 2+3+5+7+11+13+17+19+23 = 41+59 = 29+71 = 31+47+(378 mod 178)
    178 = 2+3+5+7+11+13+17+19+23 + 31+47
    200 = 2+3+5+7+11+13+17+19+23 + 41+59
    300 = 2+3+5+7+11+13+17+19+23 + 41+59 + 29+71
    378 = 2+3+5+7+11+13+17+19+23 + 41+59 + 29+71 + 31+47

    done “ch.op.eur.bri˚k
    25722 = Σs(5090) – 400 – 71
    25746 = Σs(5090) – 400 – 47
    25770 = Σs(5090) – 100 – 196883 + 196560
    f(sst) 0˚NA(!w)ru|ur”

    σ(Φ(47))^5 – Φ(47)^5 – 47^5 = 19^5 + 43^5 + 67^5

  44. Paul Vaughan says:

    25722 = Σs(5090) – s(2401) – 71
    25746 = Σs(5090) – ΣΦ(196883-196560) + Σφ(196883-196560)) – 47
    25770 = Σs(5090) – 4370 + s(4370) – 323 ; 4370 – s(4370) = s(194)

    Standish (1992) anomalitic:
    317.450267066044 = 1/(1/2/11.8627021700857-3/2/29.4701958106261+1/84.0331316671926)

  45. Paul Vaughan says:

    ˚Too *$ the conCOIIUSh’n’writeSAMicAB11y
    25674 = Σs(5090)-Σs(242)-47-average(59*59,-47*71)
    242 = 71 + 171
    714463914.954807 = beat(29.42351935,29.4235181382615) —– (Seidelmann1992,NASA’factsheet’)
    1714009140.222 = beat(29.447498973306,29.4474984673838) — (Seidelmann1992,Sidorenkov2017)
    IT$UNknown ‘weather friendly’ giants are in2cn moonSST!R UShine
    1225154850 = beat(1714009140.222,714463914.954807)
    2450309700, 4900619400, 9801238800, …
    3675464550, 7350929100, 14701858201, …
    (m)essturn wise sh!owe D-IT B(!w)(un)k(umou)
    242 = ΣΣΔ(178)-ΣΣφ(178) = average(ΣΔ(220),Σδ(220)) = ΣΦ(163)-Σφ(163)
    29.4474987203449 = harmean(29.447498973306,29.4474984673838) — (Seidelmann1992,Sidorenkov2017)
    29.4235187441307 = harmean(29.42351935,29.4235181382615) —– (Seidelmann1992,NASA’factsheet’)
    36132.1889074394 = beat(29.4474987203449,29.4235187441307)
    25674 = beat(36132.1889074394,15009.1608487337) ; 15009.1608487337 = 5482096/365.25
    142 = ΣΔ(220) = 2*71
    242 = average(142,342) = 2*11^2 = 71+171
    342 = Σδ(220) = 2*171

  46. Paul Vaughan says:

    4724 to 5256

    25761.5669315114 = beat(1.00001743371442,0.999978616353183) —- Seidelmann1992
    25768.5314808954 = harmean(25775.4997969807,25761.5669315114) —————————-
    25775.4997969807 = beat(1.00001741273101,0.999978616353183) —- Sidorenkov2017

    25773.8517155112 = beat(1.0000174152119,0.999978616353183) — Standish1992 (mediates)

    5256.6393995685 J
    5254.8227273181 S
    5258.32377697949 U
    5255.227007452 N
    5256.25286945517 = JSUN harmean
    5256.24218758401
    = beat( beat( 25808.2447032344, 25773.8517155112 / 2 ) / 6 , 2362.08778401782 )

  47. Paul Vaughan says:

    anomalistic-nodal 2320 years

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857) — anomalistic
    61.0124738503575 = slip(29.4511026866654,11.862499899747) —– nodal
    851.495746676794 = slip(61.0124738503575,19.8630730877524) — nodal
    derivation (via generalized Bollinger method)
    16.5866057119762 = axial(851.49574667679,16.9161211952138)
    100.762038265834 = slip(19.8549641949401,16.5866057119762)
    2320.03347068461 = slip(100.762038265834,16.9161211952138)

    2320.03347068424 = 1/(-7/2/11.8627021700857+17/2/29.4701958106261-12/11.862499899747+30/29.4511026866654)

    purely anomalistic 2362 years derived above
    2362.08778401334 = 1/(31/11.8627021700857-77/29.4701958106261)

    combine
    130310.598028785 = beat(2362.08778401334,2320.03347068424)
    130310.598021243 = 1/(-69/2/11.8627021700857+171/2/29.4701958106261-12/11.862499899747+30/29.4511026866654)

    compare
    130192.356944535 = beat(2362.05130068208,2319.96076275948) — NASA ‘factsheet’ tropical
    orbital invariant with harmonic means of Seidelmann & NASA ‘factsheet’ tropical:
    130476.603432851 = 1/(3/11.8565239747501-8/29.4235187441307-2/83.7474063531896+7/163.72320389271)
    130762.093817962 = beat(2361.92512664087,2320.01916295313) — Seidelmann tropical

    130704.452624679 = beat(173901.37537739,74619.9907876555) — La(2004a,2010a)average

  48. Paul Vaughan says:

    nodal-anomalistic 836 years

    9.93153654387618 = 19.8630730877524 / 2
    19.8630730877524 = beat(29.4511026866654,11.862499899747) —————– nodal
    16.912768715208 = harmean(29.4511026866654,11.862499899747) ————- nodal

    6.57189983390616 = axial( 29.4701958106261 / 2 , 11.8627021700857 ) —- anomalistic

    10.7485186386365 = beat(16.912768715208,6.57189983390616)
    130.662968425853 = slip(19.8630730877524,10.7485186386365)
    835.601801700049 = slip(130.662968425853,9.93153654387618)

    836 = 11 * ( mod(163,24) + mod(67,24) + mod(43,24) + mod(19,24) )
    836 is the smallest weird number that is also an untouchable number”

  49. Paul Vaughan says:

    JS Heart

    4.96576827193809 = 19.8630730877524 / 4
    9.93153654387618 = 19.8630730877524 / 2
    19.8630730877524 = beat(29.4511026866654,11.862499899747) —————– nodal

    2.114096089401 = 8.456384357604 / 4
    4.228192178802 = 8.456384357604 / 2
    8.456384357604 = axial(29.4511026866654,11.862499899747)
    16.912768715208 = harmean(29.4511026866654,11.862499899747) ————- nodal

    6.57189983390616 = axial( 29.4701958106261 / 2 , 11.8627021700857 ) —- anomalistic

    set up generalized Bollinger method

    10.7485186386365 = beat(16.912768715208,6.57189983390616)
    5.37425931931824 = 10.7485186386365 / 2
    2.68712965965912 = 10.7485186386365 / 4

    4.73283332391597 = axial(16.912768715208,6.57189983390616)
    2.36641666195799 = 4.73283332391597 / 2
    1.18320833097899 = 4.73283332391597 / 4

    130.662968425853 = slip(19.8630730877524,10.7485186386365)
    65.3314842129265 = slip(19.8630730877524,5.37425931931824)
    50.6802153502585 = slip(19.8630730877524,2.68712965965912)

    100.895781229192 = slip(19.8630730877524,4.73283332391597)
    50.4478906145961 = slip(19.8630730877524,2.36641666195799)
    93.4596178908348 = slip(19.8630730877524,1.18320833097899)

    derive 317, 836, 1470, 1800, 4670, 100ka, & more (some omitted for now)

    2937.90591403587 = slip(100.895781229192,16.912768715208)
    1468.95295701793 = slip(100.895781229192,8.456384357604)
    734.476478508967 = slip(100.895781229192,4.228192178802)
    734.476478508967 = slip(50.4478906145961,4.228192178802)

    1798.69939643745 = slip(93.4596178908348,8.456384357604)
    899.349698218727 = slip(93.4596178908348,4.228192178802)

    317.021047394046 = slip(100.895781229192,4.96576827193809)
    317.021047394066 = slip(93.4596178908348,19.8630730877524)

    835.601801700049 = slip(130.662968425853,9.93153654387618)
    835.601801700049 = slip(309.763646655713,130.662968425853)
    835.601801700042 = slip(225.988187105058,130.662968425853)
    835.60180170002 = slip(130.662968425853,112.994093552529)
    where
    309.763646655713 = slip(130.662968425853,19.8630730877524)
    225.988187105058 = slip(65.3314842129265,19.8630730877524)
    112.994093552529 = slip(50.6802153502585,19.8630730877524)

    4670.02054486764 = slip(1344.24031754827,130.662968425853)
    where
    1344.24031754827 = slip(130.662968425853,4.228192178802)
    1344.24031754825 = slip(144.731137488337,130.662968425853)
    144.731137488337 = slip(65.3314842129265,4.228192178802)

    198956.792259811 = slip(14755.2250793229,130.662968425853)
    99478.3961299054 = slip(14755.2250793229,65.3314842129265)
    where
    14755.2250793229 = slip(50.6802153502585,16.912768715208)

    compare sidereal vs. anomalistic
    16.9132450828034 = harmean(29.4571309198874,11.861990807677)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    99476.8155050052 = beat(16.9161211952138,16.9132450828034)
    99477 review

  50. Paul Vaughan says:

    Seidelmann (1992) sidereal earth

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

    1.00001743390371 = (1-(1/σ(σ(σ(73))))^1)^(0/1)/(1-(1/σ(σ(σ(73))))^2)^(2/2)/(1-(1/σ(σ(σ(73))))^3)^(3/3)/(1-(1/σ(σ(σ(73))))^4)^(2/4)/(1-(1/σ(σ(σ(73))))^5)^(5/5)/(1-(1/σ(σ(σ(73))))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(209))^1)^(0/1)/(1-(1/σ(209))^2)^(2/2)/(1-(1/σ(209))^3)^(3/3)/(1-(1/σ(209))^4)^(2/4)/(1-(1/σ(209))^5)^(5/5)/(1-(1/σ(209))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(47+59+71))^1)^(0/1)/(1-(1/σ(47+59+71))^2)^(2/2)/(1-(1/σ(47+59+71))^3)^(3/3)/(1-(1/σ(47+59+71))^4)^(2/4)/(1-(1/σ(47+59+71))^5)^(5/5)/(1-(1/σ(47+59+71))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(average(320,158)))^1)^(0/1)/(1-(1/σ(average(320,158)))^2)^(2/2)/(1-(1/σ(average(320,158)))^3)^(3/3)/(1-(1/σ(average(320,158)))^4)^(2/4)/(1-(1/σ(average(320,158)))^5)^(5/5)/(1-(1/σ(average(320,158)))^6)^(1/6)

    1.00001743390371 = (1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^1)^(0/1)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^2)^(2/2)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^3)^(3/3)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^4)^(2/4)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^5)^(5/5)/(1-(1/σ(average(28+163+67+43+19,10+13+18+22+37+58)))^6)^(1/6)

    keywords: Schneider, E8, monster, Ramanujan
    73: lowest prime congruent to 1 mod 24

  51. Paul Vaughan says:

    another 317 year note

    744 = σ(743)
    743.744122286576 = slip(317.021047394066,130.662968425853)
    744 = σ(240)
    240 = σ(σ(σ(73))) = σ(average(320,158)) = σ(average(28+163+67+43+19,10+13+18+22+37+58))

    examples:

    104 = d(2,1/2,58) = R(2,1/2,58) – R(1,1/2,58)
    104 = d(4,1/2,58) = R(4,1/2,58) – R(1,1/2,58)
    R(4,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π
    R(2,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π
    R(1,1/2,58) = 0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π

    744 = d(3,1/2,28) = R(3,1/2,28) – R(1,1/2,28)
    R(3,1/2,28) = 744.01187441498 = ⌊(e^√28π)^(1/3)⌉^3 – e^√28π
    R(1,1/2,28) = 0.0118744149804115 = ⌊(e^√28π)^(1/1)⌉^1 – e^√28π

    The middle argument “1/2” corresponds to “√” — i.e. raise to power 1/2; e.g. 28^(1/2) = √28

  52. Paul Vaughan says:

    744 = σ(240)
    240 =σ(209) = σ(47+59+71) = σ(average(ΣΦ(216),Σφ(216)))
    216 = 378-ΣΔ(378) = Φ(ΣΦ(323)+Σφ(323))

    1.00001743390371 = (1-(1/σ(average(ΣΦ(216),Σφ(216))))^1)^(0/1)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^2)^(2/2)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^3)^(3/3)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^4)^(2/4)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^5)^(5/5)/(1-(1/σ(average(ΣΦ(216),Σφ(216))))^6)^(1/6)

    Seidelmann (consciously? or not?) simply substituted 240 for Schneider‘s phi.

  53. Paul Vaughan says:

    Seidelmann (1992) synodic helps clarify bias structure.

    11.8619992845449 = 1/(-1/1.0920848733744+1/1.00001743301243) — note sidereal earth period
    29.4571720000365 = 1/(-1/1.03515920602327+1/1.00001743301243)

    936.955612647599 = 1/(178/11.8619992845449-442/29.4571720000365)
    1800.77215359294 = 1/(29/11.8619992845449-72/29.4571720000365)

    11.8619992845449 = beat(1.0920848733744,1.00001743301243)
    29.4571720000365 = beat(1.03515920602327,1.00001743301243)

    set up generalized Bollinger method

    19.8589101021728 = beat(29.4571720000365,11.8619992845449)
    9.92945505108639 = 19.8589101021728 / 2
    4.9647275255432 = 19.8589101021728 / 4

    16.9132604709107 = harmean(29.4571720000365,11.8619992845449)
    8.45663023545537 = axial(29.4571720000365,11.8619992845449)
    4.22831511772768 = 8.45663023545537 / 2
    2.11415755886384 = 8.45663023545537 / 4

    10.7442781148351 = beat(16.9132604709107,6.57038853131229)
    5.37213905741754 = 10.7442781148351 / 2
    2.68606952870877 = 10.7442781148351 / 4

    9.46417591360268 = harmean(16.9132604709107,6.57038853131229)
    4.73208795680134 = axial(16.9132604709107,6.57038853131229)
    2.36604397840067 = 4.73208795680134 / 2
    1.18302198920033 = 4.73208795680134 / 4

    derive 936 & 1800

    130.930052601617 = slip(19.8589101021728,10.7442781148351)
    65.4650263008087 = slip(19.8589101021728,5.37213905741754)

    506.04166179034 = slip(130.930052601617,16.9132604709107)
    703.826579261491 = slip(130.930052601617,9.92945505108639)

    1800.77215358773 = beat(703.826579261491,506.04166179034)

    351.913289630745 = slip(130.930052601617,4.9647275255432)
    936.955612639497 = slip(351.913289630745,65.4650263008087)

    3747.82245055826 = slip(130.930052601617,4.22831511772768) ; / 4 = 936.955612639565

  54. Paul Vaughan says:

    Introducing Moonshine Bias

    1.0000262476142 —- Standish (1992) anomalistic earth period
    1.00002624761586 = 30/(15/0.0745030006844627+16/0.0754402464065708-31/0.0808503463381246-1/2364963.50364963+8/7201)
    2364963.50364963 = beat(74619.9907876555,72337.575351641) — La(2004a,2010a)average

    7201: 1, 19, 379, 7201; harmean = 3.79
    7201 = 19 + ( 19 * 378 )

    7200.99967683779 = 8/(30/1.0000262476142-15/0.0745030006844627-16/0.0754402464065708+31/0.0808503463381246+1/2364963.50364963)

    7201.000000 = 8/(30/1.00002624761586-15/0.0745030006844627-16/0.0754402464065708+31/0.0808503463381246+1/2364963.50364963)
    _____________________________________________________________________________________________
    monstrous error:
    -0.000023823837 seconds ( temporal (not angular) ) per century

  55. Paul Vaughan says:

    supplementary art dove wiser
    118239.762151809 = beat(1814.75583949423,1787.32381267774)

    1800.249870433 = harmean(2364795.24303618,900.467685151888)
    1800.2498895435 = harmean(2364861.19623035,900.467685151888)
    1800.24991918585 = harmean(2364963.50364963,900.467685151888)

    scale(un)80/4
    84.021214079097 = beat(1.0120629705681,1.00001743390371)
    164.770564556546 = beat(1.00612375085558,1.00001743390371)

    111.291642790288 = harmean(164.770564556546,84.021214079097)
    111.307357343015 = harmean(164.793624044745,84.0331316671926)

    788287.065410296 = beat(111.307357343015,111.291642790288)
    2364861.19623089 = 3 * 788287.065410296

  56. Paul Vaughan says:

    Swiss bunkers watch M11

    0 = ⌊(70^2-55)/2^x-2^(2^2^2-x)*27/365.25⌉, x=0,1,2,…

    317.44 = s(s(608))*(2/(Φ+φ))^4
    317.44 = 496*(4/5)^2 = 496*16/25

    What Fairbridge said was:
    ‘[…] 317.749 years […]. A storminess record in geomorphic (that is, physical) form is preserved in a “staircase” of 184 isostatically uplifted beach lines on Hudson Bay (Fairbridge and Hillaire-Marcel” 1977, Nature. Vol. 268), which date back to more than 8,000 years. Their extraordinary regularity is duplicated in other parts of the Arctic, which denies any theory of randomness in storminess cycles. […]’

    25684 = Σs(5090) – 509 ; 509 * 59 = 30031; 30031 lowest primorial+1 not prime
    25808 = beat(beat(25746,25684),25746) = 1/(2/25746-1/25684)

  57. Paul Vaughan says:

    Phi(un)ally Perfect
    sidereal JS estimates

    x = average(Φ,φ)√√(y/496); search “why?” = 317.45026706604

    106975.540995836 = beat(1.00001743390371,1.00000808573393)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857) — anomalistic
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)

    with sense(un)a dove mercury:
    231842.576028623 = 1 / g_1
    σ(25808) = 50034; * 2 = 100068
    198928.033687742 = beat(231428.571428571,106975.540995836)
    16.9132447127788 = axial(99464.016843871,16.9161211952138)
    19.8589044939136 = beat(100068,19.8549641949401)
    14.7285651846795 = beat(19.8589044939136,8.45662235638942)
    5.93099526646578 = axial(19.8589044939136,8.45662235638942)

    19.8589044939136 = beat(29.4571303693591,11.8619905329316) — compare Standish sidereal

    4.73208318183218 = axial(16.9132447127788,6.57038170393041)
    2.36604159091609 = 4.73208318183218 / 2
    1.18302079545805 = 4.73208318183218 / 4

    100.985212897808 = slip(19.8589044939136,4.73208318183218)
    50.4926064489041 = slip(19.8589044939136,2.36604159091609)
    93.0623385488049 = slip(19.8589044939136,1.18302079545805) ~= 744/8
    ⌊744.498708390439⌉ = 744

    3455.67836213557 = slip(50.492606448904,16.9132447127788)
    enlightenin’12sense: aro(un)d‘ve 3456 sing un
    1727.83918106778 = slip(50.492606448904,8.45662235638942)

    1186.17557935921 = slip(100.985212897808,19.8589044939136)

    3455.67836213692 = slip(100.985212897808,16.9132447127788)
    3455.67836213692 = slip(50.4926064489041,16.9132447127788)
    1727.83918106846 = slip(100.985212897808,8.45662235638942)
    1727.83918106846 = slip(50.4926064489041,8.45662235638942)

    ⌊1727.83918106846⌉ = 1728
    no. won nos. weather myth or math

  58. Paul Vaughan says:

    at the Heart of Fleur Dehli nos.

    317.021047394046 = “why?”
    2876.43697242977 = beat(1.00001743390371,0.999669890283597)
    2876.43340223995 = (5+Φ)*2^9

  59. Paul Vaughan says:

    100, 496, 836, 73500

    19.8589044939136 = beat(28.8814946499969,11.7675450749242)
    9.92945224695679 = 19.8589044939136 / 2
    4.9647261234784 = 19.8589044939136 / 4

    16.7218853102037 = harmean(28.8814946499969,11.7675450749242)
    8.36094265510182 = axial(28.8814946499969,11.7675450749242)
    4.18047132755091 = 8.36094265510182 / 2
    2.09023566377546 = 8.36094265510182 / 4

    10.5902904904416 = beat(16.7218853102037,6.48390755374098)
    5.29514524522082 = 10.5902904904416 / 2
    2.64757262261041 = 10.5902904904416 / 4

    159.124845970951 = slip(19.8589044939136,10.5902904904416)

    1246.02534363448 = slip(159.124845970951,2.09023566377546); coefficient 76
    7350.70156103038 = slip(1246.02534363448,159.124845970951)

    2492.05068726896 = slip(159.124845970951,4.18047132755091)
    7350.70156103038 = slip(2492.05068726896,159.124845970951)

    79.5624229854757 = slip(19.8589044939136,5.29514524522082)
    3675.35078051519 = slip(1246.02534363448,79.5624229854757)

    3115.06335908839 = slip(79.3092165212675,4.9647261234784)
    7350.70156127398 = slip(3115.06335908839,159.124845970951)

    12460.2534363458 = slip(159.124845970951,19.8589044939136)
    30082.0055397318 = slip(12460.2534363571,4.18047132755091)
    323010.783812806 = slip(30082.0055397318,79.5624229854757); coefficient 378

    simple enough

  60. Paul Vaughan says:

    plate 000˚k (Alive in 1728 over herd “he’s just taco˚k shun”)

    quote:
    12 = 71-59 = 59-47
    3^3 = 27
    4^3 = 64
    5^3 = 125; 125+64+27 = 216; 216000 = 125*64*27
    6^3 = 216

    supplement:

    Φ(25771) = 25770
    σ(σ(25770)) = 216216

    σ(σ(25770)) = (3^3*4^3*5^3)+(6^3)
    σ(σ(25770)) = (3^3*4^3*5^3)+(3^3+4^3+5^3)

    1 / g_3 =
    74619.9907876555 — La2011 Table 6 La2004a
    74619.9907876555 — La2011 Table 6 La2010a
    74621: σ(σ(74621)) = σ(σ(Φ(25771))) = σ(σ(25770)) = 216216
    74626.0277273697 — La2011 Table 5

  61. Paul Vaughan says:

    with NASA ‘factsheet’ synodic
    1.00001743356471 = 1/(1/100/(φφ/(1/11.8629550321199+1/29.4600280504908)+1/(1/11.8629550321199-1/29.4600280504908))+1/√5/√√(1/(1/11.8629550321199-1/29.4600280504908)/496))

  62. Paul Vaughan says:

    blaring Soundgarden’s “black hole sun”
    speedboat cruised at sunset
    past an island up the inlet between the mountains
    starboard featured nice blue pentagon design

    understand how whoever came with that for the ‘factsheet’?

    above used right side of illustration
    x = average(Φ,φ)√√(y/496)

    left side:
    z = 2*average(Φ,φ)√√(u/496) = (√5)*(√√(u/496))
    ‘factsheet’ synodic: substitute u =
    19.8602908360448 = beat(29.4600280504908,11.8629550321199)

    4200.19506151223 = beat(1.00025558289712,1.00001743371442)
    underscore this: try same calculation on EVERY other parameter list ever featured at the talkshop — you won’t find ANYTHING even remotely close — TUNING is the obvious interpretation

    compare:
    8.45735138020641 = axial(29.4600280504908,11.8629550321199)
    22.1416333681812 = φφ * 8.45735138020641
    42.001924204226 = 22.1416333681812 + 19.8602908360448 —- review
    4200.1924204226 = 100 * 42.001924204226

    a little algebra, arrive (it isn’t really curious) at a very precise estimate of their (lunisolar-biased) sidereal earth year-length

    recommendation: stop the sanctions now (PLEASE: don’t delay)

  63. Paul Vaughan says:

    Mayan Sun

    review — notes shared a few years ago now connected with k & s_3
    25684.9315068493 = 360*60*60/50.4576
    68756.6342763388 = beat(41001.6165713381,25684.9315068493)
    68756.6342763387 = 1/(1/360/60/60*50.4576-1/2^9/5^3/13/(5256.63940169013)*(44.2784629967671)*(73.0002008969005)*11*3)
    41001.6165713381 = 2^9*5^3*13*(5256.63940169013)/( 44.2784629967671)/(73.0002008969005)/11/3
    5125.20207141727 = 41001.6165713381 / 8
    394.246313185944 = 5125.20207141727 / 13
    143998.465891166 = 394.246313185944 * 365.25

    NASA ‘factsheet’ synodic
    carefully scrutinize 365.256 days in concert with scaling featured in last few comments
    11.8631499061245 = beat(1.09207392197125,1.00001642710472)
    29.4605119934568 = beat(1.03515400410678,1.00001642710472)
    83.9387085475763 = beat(1.01207392197125,1.00001642710472)
    164.501359353944 = beat(1.00613278576318,1.00001642710472)
    68753.7838334262 = 1/(-2067/11.8631499061245+6233/29.4605119934568-2075/83.9387085475763-2075/164.501359353944)
    41002.6302294426 = beat(68753.78398,25684.9315068493)
    5125.32877868033 = 41002.6302294426 / 8
    394.256059898487 = 5125.32877868033 / 13
    144002.025877922 = 394.256059898487 * 365.25

    baktun balance with Standish (1992) long-duration sidreal earth year-length
    11.8632889636452 = beat(1.09207392197125,1.0000174152119)
    29.4613695891215 = beat(1.03515400410678,1.0000174152119)
    83.9456708028269 = beat(1.01207392197125,1.0000174152119)
    164.528101664775 = beat(1.00613278576318,1.0000174152119)
    5125.26068671143 = 1/(32.5/11.8632889636452-81.5/29.4613695891215+1.5/83.9456708028269+1.5/164.528101664775)
    5125.26068671143 = 2/(65/11.8632889636452-163/29.4613695891215+3/83.9456708028269+3/164.528101664775)
    68755.3156707909 = beat(41002.0854936914,25684.9315068493)
    41002.0854936914 = 5125.26068671143 * 8
    5125.26068671143 = 41002.0854936914 / 8
    394.250822054725 = 5125.26068671143 / 13
    144000.112755488 = 394.250822054725 * 365.25

    generalized Bollinger (1952) method used to derive all of the preceding

  64. Paul Vaughan says:

    typo near end of lunisolar 1800 comment

    supplementary
    68952.9911154182 = slip(1800.93537030388,0.999978616353183) — M&S unrounded
    68961.1021023274 = slip(1800.93537030388,0.999978614647502) —— M&S rounded

  65. Paul Vaughan says:

    The Real Thing

    generalized Bollinger setup

    NASA ‘factsheet’ “Perihelion (10^6 km)”
    18.5132744565067 = beat(27.1859355652811,11.0133345026533)
    9.25663722825335 = 18.5132744565067 / 2
    4.62831861412668 = 18.5132744565067 / 4
    15.6761007011677 = harmean(27.1859355652811,11.0133345026533)
    7.83805035058385 = axial(27.1859355652811,11.0133345026533)
    3.91902517529192 = 7.83805035058385 / 2
    1.95951258764596 = 7.83805035058385 / 4

    crossed with Standish (1992) anomalistic
    11.3158491444058 = beat(15.6761007011677,6.57189983390616)
    5.65792457220292 = 11.3158491444058 / 2
    2.82896228610146 = 11.3158491444058 / 4
    9.26121549052347 = harmean(15.6761007011677,6.57189983390616)
    4.63060774526174 = axial(15.6761007011677,6.57189983390616)
    2.31530387263087 = 4.63060774526174 / 2
    1.15765193631543 = 4.63060774526174 / 4

    hierarchy top levels
    50.8673777758235 = slip(18.5132744565067,11.3158491444058)
    68.039402479501 = slip(18.5132744565067,5.65792457220292)
    40.616453731329 = slip(18.5132744565067,2.82896228610146)

    9362.47281505814 = slip(18.5132744565067,4.63060774526174)
    4681.23640752907 = slip(18.5132744565067,2.31530387263087)
    2340.61820376454 = slip(18.5132744565067,1.15765193631543)

    33052.6407001566 = slip(9362.47281505814,18.5132744565067)

  66. Paul Vaughan says:

    Standish anomalistic JS
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    crossed with nasa ‘factsheet’ synodic (using Standish long-duration earth sidereal)
    10.7452986346853 = beat(16.9161211952138,6.57120187976141)
    130.437274099443 = slip(19.8549641949401,10.7452986346853)
    451.069113778482 = slip(130.437274099443,16.9161211952138)
    984.586552944021 = slip(451.069113778482,130.437274099443)
    980.005077317815 = beat(984.586552944021,491.145236537217)
    73500.3807988361 = 980.005077317815 * 75
    36750.1903994181 = 73500.3807988361 / 2

  67. Paul Vaughan says:

    33052.6407001566 ; / 8 / √φ = 5255.45105486596 ; / 8 / √φ = 835.629626106117
    36133.4834429326 = beat(29.4474984673838,29.42351935)
    36133.4519209772 = axial( 500 * 835.629626106117 , Φ * 64000 )
    36133.2446397442 = axial( 500 * 835.546575435627 , Φ * 64000.2003306117 )

    11.8619906635942 = axial( 250 * 835.629626106117 , 11.8627021700857 )
    11.861990807677 — compare Standish sidereal

  68. Paul Vaughan says:

    JS Seidelmann short-duration
    19.8650360864628 = beat(29.4474984673838,11.8626151546089)
    9.93251804323141 = 19.8650360864628 / 2
    4.9662590216157 = 19.8650360864628 / 4
    16.9122914926352 = harmean(29.4474984673838,11.8626151546089)
    8.4561457463176 = axial(29.4474984673838,11.8626151546089)
    4.2280728731588 = 8.4561457463176 / 2
    2.1140364365794 = 8.4561457463176 / 4

    crossed with Standish anomalistic
    10.7487113950462 = beat(16.9122914926352,6.57189983390616)
    5.37435569752312 = 10.7487113950462 / 2
    2.68717784876156 = 10.7487113950462 / 4
    9.46559190444059 = harmean(16.9122914926352,6.57189983390616)
    4.73279595222029 = axial(16.9122914926352,6.57189983390616)
    2.36639797611015 = 4.73279595222029 / 2
    1.18319898805507 = 4.73279595222029 / 4

    generalized Bollinger top-level
    130.804508068333 = slip(19.8650360864628,10.7487113950462)
    65.4022540341667 = slip(19.8650360864628,5.37435569752312)
    50.6080139018586 = slip(19.8650360864628,2.68717784876156)

    100.676696558682 = slip(19.8650360864628,4.73279595222029)
    50.3383482793411 = slip(19.8650360864628,2.36639797611015)
    94.2634877779342 = slip(19.8650360864628,1.18319898805507)

    ~ 2 * 1470 :
    2940.22366441706 = slip(639.79104597475,65.4022540341667)
    639.79104597475 = slip(94.2634877779342,8.4561457463176)

    astronomical delight
    23098.5346098618 = slip(2081.08715717028,130.804508068333) ~= 23.1 ka
    2081.08715717028 = slip(130.804508068333,4.2280728731588)

    concise lunisolar review
    29976.553895592 = beat(179.333323110834,178.266850068779) — M&S rounded tropical
    29971.9562539596 = beat(179.333487684639,178.266850068779) — M&S unrounded tropical
    contains
    29972.4308734593 = slip(6642.042967895,130.804508068333)
    6642.042967895 = slip(50.6080139018586,16.9122914926352)
    50.6080139018586 = slip(19.8650360864628,2.68717784876156)

  69. David A says:

    Your foray into orbital resonance is fascinating. To even understand orbital resonance I found this video https://youtu.be/Qyn64b4LNJ0 very explanatory to me. You all may enjoy it, not for what you already know, but for the last third or so, where the video gets into converting orbital resonance into sound. Very interesting and cool.

  70. oldbrew says:

    David A – in reality there are no *exact* orbital resonances that we know of. The true resonances are synodic, i.e. based on periods when two bodies are in line with the body they’re orbiting.

    However the Galilean moons of Jupiter are very close to an exact orbital resonance, but see here:

    Why Phi? – the resonance of Jupiter’s Galilean moons

  71. Paul Vaughan says:

    David A: reference framing & sampling/aggregation biases (not resonance) have been the more memorable exploration spices. (Look for “anomalistic periods” on the net, find little, wonder why.)

  72. Paul Vaughan says:

    no. fancy policy

    132942.038841156 = slip(2493.45557752384,50.8673777758235)
    2493.45557752384 = slip(50.8673777758235,3.91902517529192)

    anomalistic purpose?

    132942.038841155 = beat(2361.4060221632,2320.1931882465)
    2340.61820376454 = harmean(2361.4060221632,2320.1931882465)

    Dow

    non-Mayan


    pleas

  73. Paul Vaughan says:

    28.8 & 29.8 ka note

    background

    28861.8479491577 = slip(521.471474393677,65.3314842129265)
    521.471474393677 = slip(93.4596178908348,4.96576827193809)

    1268.08418957619 = slip(100.895781229192,19.8630730877524)
    59572.1478647727 = slip(1268.08418957619,50.6802153502585) ; / 2 =
    29786.0739323864

    image disappeared from here

  74. Paul Vaughan says:

    Vert

    anomalistic
    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    9.92748209747005 = 19.8549641949401 / 2
    4.96374104873503 = 19.8549641949401 / 4
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    8.45806059760692 = axial(29.4701958106261,11.8627021700857)
    4.22903029880346 = 8.45806059760692 / 2
    2.11451514940173 = 8.45806059760692 / 4

    cross with nodal
    10.7419198566295 = beat(16.9161211952138,6.56993811757712)
    5.37095992831474 = 10.7419198566295 / 2
    2.68547996415737 = 10.7419198566295 / 4
    9.46415641396783 = harmean(16.9161211952138,6.56993811757712)
    4.73207820698391 = axial(16.9161211952138,6.56993811757712)
    2.36603910349196 = 4.73207820698391 / 2
    1.18301955174598 = 4.73207820698391 / 4

    top level
    130.937221254586 = slip(19.8549641949401,10.7419198566295)
    65.4686106272931 = slip(19.8549641949401,5.37095992831474)
    50.4636420418041 = slip(19.8549641949401,2.68547996415737)

    101.392224425295 = slip(19.8549641949401,4.73207820698391)
    50.6961122126475 = slip(19.8549641949401,2.36603910349196)
    91.6213954766362 = slip(19.8549641949401,1.18301955174598)

    so what?
    323.049985568367 = slip(130.937221254586,19.8549641949401)
    220.179370109776 = slip(65.4686106272931,19.8549641949401)
    110.089685054889 = slip(50.4636420418041,19.8549641949401)
    504.343543575558 = slip(130.937221254586,16.9161211952138)

    no. doubt:
    2998.18929479554 = slip(50.4636420418041,16.9161211952138)
    1499.09464739777 = slip(50.4636420418041,8.45806059760692)

    86900.0492950508 = slip(504.343543575558,50.4636420418041)
    86900 = 11 * 25 * 316

  75. Paul Vaughan says:

    con$hiver˚T review:

  76. Paul Vaughan says:

    JEV nodal-anomalistic (casual intro)

    nodal

    320.347986973937 = slip(44.2875117414503,0.761762061330659)
    320.347986973937 = slip(22.1437558707251,0.761762061330659)
    23.7880850683634 = slip(11.0718779353626,0.761762061330659) ———-

    160.173993486969 = slip(44.2875117414503,0.38088103066533)
    160.173993486969 = slip(22.1437558707251,0.38088103066533)
    160.173993486969 = slip(11.0718779353626,0.38088103066533)

    99.0755539755134 = slip(44.2875117414503,0.190440515332665) ———-
    80.0869967434843 = slip(22.1437558707251,0.190440515332665)
    80.0869967434843 = slip(11.0718779353626,0.190440515332665)

    anomalistic

    575.799247511725 = slip(44.2411450188424,0.761769224080824)
    575.799247511725 = slip(22.1205725094212,0.761769224080824)
    23.0043337056467 = slip(11.0602862547106,0.761769224080824) ———–

    287.899623755862 = slip(44.2411450188424,0.380884612040412)
    287.899623755862 = slip(22.1205725094212,0.380884612040412)
    287.899623755862 = slip(11.0602862547106,0.380884612040412)

    143.949811877931 = slip(44.2411450188424,0.190442306020206)
    143.949811877931 = slip(22.1205725094212,0.190442306020206)
    143.949811877931 = slip(11.0602862547106,0.190442306020206)

    reorganizing

    320 = 28+163+67+43+19 (the 744 levels)
    320.347986973937 = slip(44.2875117414503,0.761762061330659) — n
    575.799247511725 = slip(44.2411450188424,0.761769224080824) — a
    576 = 320+256 = 28+163+67+43+19 + 2^8 = 4 * 12^2

    320.347986973937 = slip(22.1437558707251,0.761762061330659) — n
    575.799247511725 = slip(22.1205725094212,0.761769224080824) — a

    23.7880850683634 = slip(11.0718779353626,0.761762061330659) — n
    23.0043337056467 = slip(11.0602862547106,0.761769224080824) — a

    160.173993486969 = slip(44.2875117414503,0.38088103066533) — n
    287.899623755862 = slip(44.2411450188424,0.380884612040412) — a
    288 = 160+128 = 2 * 12^2

    160.173993486969 = slip(22.1437558707251,0.38088103066533) — n
    287.899623755862 = slip(22.1205725094212,0.380884612040412) — a

    160.173993486969 = slip(11.0718779353626,0.38088103066533) — n
    287.899623755862 = slip(11.0602862547106,0.380884612040412) — a

    99 = 163-64 —————————————————————————————— note well
    99.0755539755134 = slip(44.2875117414503,0.190440515332665) — n
    143.949811877931 = slip(44.2411450188424,0.190442306020206) — a

    80 = 144-64
    80.0869967434843 = slip(22.1437558707251,0.190440515332665) — n
    143.949811877931 = slip(22.1205725094212,0.190442306020206) — a
    144 = 80+64

    80.0869967434843 = slip(11.0718779353626,0.190440515332665) — n
    143.949811877931 = slip(11.0602862547106,0.190442306020206) — a
    144 = 59*59 – 47*71 = 12^2

  77. Paul Vaughan says:

    Tortoise

    576 = 24^2
    23 = 99 – 76
    “Experts” misled us severely last decade.
    76 = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24 = 19 + 19 + 19 + 19

    JEV nodal

    417.865531943236 = slip(160.173993486969,44.2875117414503)
    417.865531943179 = slip(99.0755539755134,44.2875117414503) ————
    417.865531943236 = slip(80.0869967434843,44.2875117414503)

    208.93276597159 = slip(99.0755539755134,22.1437558707251) ————-
    208.932765971618 = slip(80.0869967434843,22.1437558707251)

    compare with

    JEV anomalistic

    417.781912370365 = slip(49.480945018333,44.2411450188424)
    208.890956185182 = slip(49.480945018333,22.1205725094212)
    104.445478092591 = slip(49.480945018333,11.0602862547106)

    46 = 209 – 163 = 2 * 23
    σ(Φ(47))^5 – Φ(47)^5 – 47^5 = 19^5 + 43^5 + 67^5

    835.563824740778 = slip(6.84967828238651,0.761769224080824)
    417.781912370389 = slip(6.84967828238651,0.380884612040412)
    208.890956185194 = slip(6.84967828238651,0.190442306020206)
    104.445478092597 = slip(6.84967828238651,0.095221153010103)

    Study carefully:

    2096.44010053095 = slip(135.584676517528,44.2411450188424)
    135.584676517528 = slip(44.2411450188424,1.59867106414771)

  78. Paul Vaughan says:

    supplementary
    49.480945018333 = slip(6.84967828238651,0.399667766036927)

    anomalistic
    835.56382473927 = 1/(-14/0.615197860179071+22/1.0000262476142+9/11.8627021700857)

    nodal
    835.731063859634 = 4/(-1387/0.615194395759546+2333/1.00001071395229-930/11.862499899747)

    derive from:

    anomalistic

    1.59867106414771 = beat(1.0000262476142,0.615197860179071)
    0.799335532073854 = 1.59867106414771 / 2
    0.399667766036927 = 1.59867106414771 / 4
    0.761769224080824 = harmean(1.0000262476142,0.615197860179071)
    0.380884612040412 = axial(1.0000262476142,0.615197860179071)
    0.190442306020206 = 0.380884612040412 / 2
    0.095221153010103 = 0.380884612040412 / 4

    0.814043420635227 = beat(11.8627021700857,0.761769224080824)
    0.407021710317613 = 0.814043420635227 / 2
    0.203510855158807 = 0.814043420635227 / 4
    1.43160709790728 = harmean(11.8627021700857,0.761769224080824)
    0.715803548953639 = axial(11.8627021700857,0.761769224080824)
    0.35790177447682 = 0.715803548953639 / 2
    0.17895088723841 = 0.715803548953639 / 4

    44.2411450188424 = slip(1.59867106414771,0.814043420635227)
    22.1205725094212 = slip(1.59867106414771,0.407021710317613)
    11.0602862547106 = slip(1.59867106414771,0.203510855158807)

    6.84967828238651 = slip(1.59867106414771,0.715803548953639)
    3.42483914119326 = slip(1.59867106414771,0.35790177447682)
    24.0670906604157 = slip(1.59867106414771,0.17895088723841)

    nodal

    1.59868736807262 = beat(1.00001071395229,0.615194395759546)
    0.799343684036311 = 1.59868736807262 / 2
    0.399671842018155 = 1.59868736807262 / 4
    0.761762061330659 = harmean(1.00001071395229,0.615194395759546)
    0.38088103066533 = axial(1.00001071395229,0.615194395759546)
    0.190440515332665 = 0.38088103066533 / 2
    0.0952202576663324 = 0.38088103066533 / 4

    0.81403619360271 = beat(11.862499899747,0.761762061330659)
    0.407018096801355 = 0.81403619360271 / 2
    0.203509048400678 = 0.81403619360271 / 4
    1.43159297613223 = harmean(11.862499899747,0.761762061330659)
    0.715796488066113 = axial(11.862499899747,0.761762061330659)
    0.357898244033057 = 0.715796488066113 / 2
    0.178949122016528 = 0.715796488066113 / 4

    44.2875117414503 = slip(1.59868736807262,0.81403619360271)
    22.1437558707251 = slip(1.59868736807262,0.407018096801355)
    11.0718779353626 = slip(1.59868736807262,0.203509048400678)

    6.84843333335951 = slip(1.59868736807262,0.715796488066113)
    3.42421666667975 = slip(1.59868736807262,0.357898244033057)
    24.1324516929722 = slip(1.59868736807262,0.178949122016528)

  79. Paul Vaughan says:

    sh: most replayed time index? 209
    99476.8446931352 = beat(130901699.437495,99401.3061146969)
    “…wwwo˚k T˚he lline like Can. edge? yep…”

  80. Paul Vaughan says:

    Call off ice$UNhhowe???0vertThe

    Horizon$1929.72222222222 (in sidereal hindsight NA!SA)
    99401.3061147312 = 2/(1/11.8619848807702+1/29.4571542179636-1/11.8627021700857-1/29.4701958106261)

    Sidellmann (1992) tropical wwwithconvert$yen
    25761.5669315114 = beat(1.00001743371442,0.999978616353183) give$ :
    99438.5571830642 = 2/(1/11.8619843895747+1/29.4571637875065-1/11.8627021700857-1/29.4701958106261)

    11.8619855226385 = beat(25746.362539063,11.8565229295003)
    29.4571824642891 = beat(25746.362539063,29.4235181382615)
    223486.379079769 = beat(11.8626151546089,11.8619855226385)
    89574.6192358693 = beat(29.4571824642891,29.4474984673838)
    63945.0695493213 = axial(223486.379079769,89574.6192358693)
    99438.3330129135 = harmean(111743.189539885,89574.6192358693)

    llunisolr biasXbias
    11.8619844613515 = 1/(-1/0.999978616353183+1/1.00001741273101-1/27500000+1/11.8565229295003)
    29.4571759194281 = 1/(-1/0.999978616353183+1/1.00001741273101-1/27500000+1/29.4235181382615)
    223110.291681529 = beat(11.8626151546089,11.8619844613515)
    89635.1786377776 = beat(29.457175919428,29.4474984673838)
    63945.0695493126 = axial(223110.291681529,89635.1786377776)
    99401.0566990933 = harmean(111555.145840765,89635.1786377776)

    catchSSTand!sh(1992)fi$[hh]calefidereal
    99476.8155050703 = 2/(1/11.861990807677+1/29.4571309198874-1/11.8627021700857-1/29.4701958106261)

    1.00001743390371 = (1-(1/σ(209))^1)^(0/1)/(1-(1/σ(209))^2)^(2/2)/(1-(1/σ(209))^3)^(3/3)/(1-(1/σ(209))^4)^(2/4)/(1-(1/σ(209))^5)^(5/5)/(1-(1/σ(209))^6)^(1/6)
    25761.4413131157 = beat(1.00001743390371,0.999978616353183)
    11.8619833699953 = beat(25761.4413131157,11.8565239747501)
    29.4571633444961 = beat(25761.4413131157,29.4235187441307)
    16.9132428669975 = harmean(29.4571633444961,11.8619833699953)
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    99400.2228512541 = beat(16.9161211952138,16.9132428669975)

    $0 few Can˚T[hh]ellweather MI = Thor math f(ll0˚CDownTh(y/n)c$yen)$T˚ache99.9999per(inno$)centCOllatorwell

  81. Paul Vaughan says:

    ˚T˚win blew
    ˚T˚won Vert “on”
    “ain’t no cause
    5256.07266843706 = 4/(1/11.8627021700857-6/84.0331316671926-2/164.793624044745) — anomalistic
    “just abattery 4/hire” – easyDC “load up Eur. Can. on”

  82. Paul Vaughan says:

    ain’t Talkin’ (Down!)

    “more than N’folk song”
    2432.13579731858 = 2/(-1/29.4701958106261+7/84.0331316671926-8/164.793624044745)
    “$hh!keep” sayin’: Eur. IT
    1216.06789865929 = 1/(-1/29.4701958106261+7/84.0331316671926-8/164.793624044745) — anomalistic
    “give allITll˚K˚!˚C˚K˚ withh eur. fine”
    608.033949329645 = 1/(-2/29.4701958106261+14/84.0331316671926-16/164.793624044745)
    “V!$hh!UShypnoCsis”

    “$hh!achhe earn IT BRI˚C wall”

  83. Paul Vaughan says:

    ‘factsheet’ 41k ace:
    “Wrong_2Won˚T!mowwR˚Than a foe 11˚K s_on_g” – metR˚Ck
    41000.1976938051 = harmean(101554.605384649,25684.9315068493)
    406218.421537452 = 16/(-7/11.8619822039699+5/29.4571389459274+7/11.8627021700857-5/29.4701958106261)
    101554.605384363 = 4/(-7/11.8619822039699+5/29.4571389459274+7/11.8627021700857-5/29.4701958106261)
    buy us anomalistic sam pull period

    NA!SAhherdin=mowwR0˚CO2revverse
    68760.6246393283 = beat(41000.1976938051,25684.9315068493)
    C˚0llDCllear
    25684.888118931 = beat(25808.1319319395,12873.1812695315)
    25808.1319319395 = 1/(-1/83.7474058863792+2/163.723203285421+1/84.0331316671926-2/164.793624044745)
    412930.110911032 = 16/(-1/83.7474058863792+2/163.723203285421+1/84.0331316671926-2/164.793624044745)

  84. Paul Vaughan says:

    JEV nodal setup listed here

    That also leads to:
    702.588969079652 = slip(6.84843333335951,0.761762061330659)
    351.294484539826 = slip(6.84843333335951,0.38088103066533)
    175.647242269913 = slip(6.84843333335951,0.190440515332665)
    87.8236211349565 = slip(6.84843333335951,0.0952202576663324)
    alternately:
    13.696866666719 = slip(1.59868736807262,1.43159297613223)
    702.588969080693 = slip(13.696866666719,0.761762061330659)
    351.294484540347 = slip(13.696866666719,0.38088103066533)
    175.647242270173 = slip(13.696866666719,0.190440515332665)
    87.8236211350867 = slip(13.696866666719,0.0952202576663324)

    Why the systematically-structured sampling bias in the short-duration models was not acknowledged early in “climate discussion” is the crUShin’mystery.

  85. Paul Vaughan says:

    using this
    345.720408200766 = slip(130.937221254586,4.96374104873503)
    345.720408200718 = 1/(26/11.862499899747+52/29.4511026866654-61/11.8627021700857+35/29.4701958106261)
    345 review

  86. Paul Vaughan says:

    figure 7hherd dog in0˚Kin awe “what ruff?”
    691.440816401436 = 1/(13/11.862499899747+26/29.4511026866654-30.5/11.8627021700857+17.5/29.4701958106261)
    691.440816401531 = slip(130.937221254586,9.92748209747005)
    using this
    2998.18929479478 = 2/(24/11.862499899747+48/29.4511026866654-55/11.8627021700857+29/29.4701958106261)
    1499.09464739739 = 1/(24/11.862499899747+48/29.4511026866654-55/11.8627021700857+29/29.4701958106261)

  87. Paul Vaughan says:

    supplementary notes

    345.720408200718 = 1/(26/11.862499899747+52/29.4511026866654-61/11.8627021700857+35/29.4701958106261)
    691.440816401436 = 2/(26/11.862499899747+52/29.4511026866654-61/11.8627021700857+35/29.4701958106261)

    691.440816401528 = slip(323.049985568367,130.937221254586)
    691.440816401528 = slip(220.179370109776,130.937221254586)

    691.44081640155 = slip(130.937221254586,110.089685054889)
    345.720408200775 = slip(110.089685054889,65.4686106272931)

    345.720408200766 = slip(130.937221254586,4.96374104873503)
    691.440816401531 = slip(130.937221254586,9.92748209747005)

    323.049985568367 = slip(130.937221254586,19.8549641949401)

    504.343543575558 = slip(130.937221254586,16.9161211952138)
    504.343543575558 = slip(65.4686106272931,16.9161211952138)

    220.179370109776 = slip(65.4686106272931,19.8549641949401)
    220.128528176077 = slip(91.6213954766362,16.9161211952138)

    323 = 196883 – 196560
    504 = 220 + s(220) ; 220 = s(s(220))

    504.343543575571 = 2/(-16/11.862499899747-32/29.4511026866654+39/11.8627021700857-25/29.4701958106261)

    220.179370109781 = 1/(6/11.862499899747+12/29.4511026866654-14/11.8627021700857+8/29.4701958106261)

    323.049985568356 = 2/(-14/11.862499899747-28/29.4511026866654+33/11.8627021700857-19/29.4701958106261)

  88. Paul Vaughan says:

    distinction (from 220.17937010978)
    220.128528176035 = 2/(40/11.862499899747+80/29.4511026866654-149/11.8627021700857+191/29.4701958106261)

  89. Paul Vaughan says:

    AmereR0˚CO$pereRllX!

    “Perihelion (10^6 km)”
    58.0330166444241 = 1/(1/11.0133345026533-2/27.1859355652811)

    152.138222152323 = 1/(1/78.4417377748537-1/161.934271741915)
    /
    6.08396577991655 = 1/(1/11.0133345026533+2/27.1859355652811)
    ~25 = 5^2 = 317-163-67-43-19

    bias SAM+pu+ll+in’SSTop! goes withhout sayUN:JS˚T chll ouT˚

  90. Paul Vaughan says:

    Table 8 Vote$$well

    “BR˚tmI Pence$$sol.
    ‘Gov.ME$$sum˚Thhun˚T00‘wwrite Don”

    e+v=47+71=2*59
    e-v=58*2^3
    e=58*2^3+v
    “$wwheat $wwheat Wall Dough”
    58*2^3+v+v=47+71
    2v=47+71-58*2^3
    e-173=47+71
    e=47+71+173
    VA’n’hale unhhothh˚TeaChR “NOAA bout this $Ch˚˚ll”
    160.17399348638 = 1/(-173/0.615194395759546+291/1.00001071395229-116/11.862499899747)
    287.899623750069 = 1/(-173/0.615197860179071+291/1.0000262476142-116/11.8627021700857)
    D!monde Ave.: “$IT Down Wall Dough”

  91. Paul Vaughan says:

    M!55 UNllink (well flower)

    lofty weather dog’n’dig note orrery:
    132891.64579987 = 2/(1/29.4571309198874-1/29.4701958106261)

    2-PEace˚TypoV How$ensoch!ustech

    25808 = Σs(5090) – 7*55
    25808 = Σs(5090) – 7*(28^2-27^2)
    25808 = Σs(5090) – 7*(1^2+2^2+3^2+4^2+5^2)

    25808 = Σs(5090) – 7*(378-323)
    25808 = Σs(5090) – 7*(378+196560-196883)
    25808 = Σs(5090) – 7*11*√(317-163-67-43-19)

    5 = √(317-163-67-43-19)
    55 = 11*√(317-163-67-43-19)
    378 = 323+11*√(317-163-67-43-19)

    notion ignore˚Folk˚CUS$

  92. Paul Vaughan says:

    4266 (or 4267) years

    curiosity noted before

    NASA Horizons 1929.72222222222 sidereal
    60.9469869005405 = slip(29.4571542179636,11.8619848807702) ; * 70 =
    4266.28908303783

    standish sidereal long-duration
    4267.83999767789 = slip(164.786005834669,84.01495797691)
    standish sidereal SHORT-duration
    4266.98089346112 = slip(164.790305314929,84.0175261973943)

    4266.96721287228 = 1/(-10/11.8627021700857-10/29.4701958106261+10/11.862499899747+10/29.4511026866654)
    Standish anomalistic with nodal — insight from CAREFUL study of “Chandler Diversity”

    same line of inquiry clarifies:
    173927.260141636 = 1/(-4/11.8627021700857+4/11.862499899747)
    173901.37537739 = 1 / g_2 —– La(2004a,2010a)average

  93. Paul Vaughan says:

    automating discovery of 173 (or 174) ka (see preceding comment)

    anomalistic-nodal combos arising with generalized Bollinger

    4.73181582715489 = axial(16.912768715208,6.56993811757712)
    4.73207820698391 = axial(16.9161211952138,6.56993811757712)
    4.73283332391597 = axial(16.912768715208,6.57189983390616)
    4.73309581660092 = axial(16.9161211952138,6.57189983390616)
    quarter-beats
    4374.2817110478 = beat(1.18327395415023,1.18295395678872)
    5502.44887404215 = beat(1.18320833097899,1.18295395678872)
    5502.44887404534 = beat(1.18327395415023,1.18301955174598)
    7414.79116630378 = beat(1.18320833097899,1.18301955174598)
    21334.8360643294 = beat(1.18301955174598,1.18295395678872)
    21334.8360643774 = beat(1.18327395415023,1.18320833097899)

    21334.8360643294 = slip(5502.44887404534,4374.2817110478)
    21334.8360643774 = slip(7414.79116630378,5502.44887404534)
    24317.7838264046 = slip(7414.79116630378,4374.2817110478)

    173927.260144754 = slip(21334.8360643774,4374.2817110478)
    173927.260141567 = slip(21334.8360643774,5502.44887404534)
    173927.260141568 = slip(21334.8360643774,7414.79116630378)
    173927.260138382 = slip(24317.7838264046,21334.8360643774)

    alternately – via other side
    10.7419198566295 = beat(16.9161211952138,6.56993811757712)
    10.7432721445968 = beat(16.912768715208,6.56993811757712)
    10.7471650296459 = beat(16.9161211952138,6.57189983390616)
    10.7485186386365 = beat(16.912768715208,6.57189983390616)
    1/4 beats
    4374.28171104601 = beat(2.68712965965912,2.68547996415737)
    5502.44887404212 = beat(2.68679125741149,2.68547996415737)
    5502.44887404128 = beat(2.68712965965912,2.68581803614919)
    7414.79116630151 = beat(2.68679125741149,2.68581803614919)
    21334.8360643478 = beat(2.68581803614919,2.68547996415737)
    21334.8360643352 = beat(2.68712965965912,2.68679125741149)

    another way to look at it – building blocks
    pure
    1.18295395678872 = 1/(+6/11.862499899747+10/29.4511026866654)
    1.18327395415023 = 1/(6/11.8627021700857+10/29.4701958106261)
    crosses
    1.18301955174598 = 1/(2/11.8627021700857+2/29.4701958106261+4/11.862499899747+8/29.4511026866654)
    1.18320833097899 = 1/(4/11.8627021700857+8/29.4701958106261+2/11.862499899747+2/29.4511026866654)

    simple subtraction
    4374.28171104854 = 1/(-6/11.8627021700857-10/29.4701958106261+6/11.862499899747+10/29.4511026866654)
    5502.44887404102 = 1/(-4/11.8627021700857-8/29.4701958106261+4/11.862499899747+8/29.4511026866654)
    7414.7911662994 = 1/(-2/11.8627021700857-6/29.4701958106261+2/11.862499899747+6/29.4511026866654)
    21334.8360643551 = 1/(-2/11.8627021700857-2/29.4701958106261+2/11.862499899747+2/29.4511026866654)

    note with care:
    4266.96721287228 = 1/(-10/11.8627021700857-10/29.4701958106261+10/11.862499899747+10/29.4511026866654)
    arises in
    173927.260144754 = slip(21334.8360643774,4374.2817110478)
    173927.260141636 = 1/(-4/11.8627021700857+4/11.862499899747)

    so that clarifies another source of systematic bias in the short-duration models that attracted attention long ago but remained mysterious at the time (no longer so now)

  94. Paul Vaughan says:

    “Does A NY buddy no. how the SSTory really goes?”

    1.59868960462765 = beat(1.0000174152119,0.615197263077614)
    0.799344802313826 = 1.59868960462765 / 2

    0.814043321555892 = beat(11.861990807677,0.761766203759125)
    0.407021660777946 = 0.814043321555892 / 2
    0.203510830388973 = 0.814043321555892 / 4

    44.2698538014441 = slip(1.59868960462765,0.814043321555892)
    “0=range(C+R)-USh!…”
    22.1349269007221 = slip(1.59868960462765,0.407021660777946)
    “…˚Take IT away buoys”
    11.067463450361 = slip(1.59868960462765,0.203510830388973)

    143.424905399078 = slip(44.2698538014441,1.59868960462765)
    143.424905399078 = slip(22.1349269007221,1.59868960462765)
    143.424905399078 = slip(11.067463450361,1.59868960462765)
    71.7124526995389 = slip(22.1349269007221,0.799344802313826)
    71.7124526995389 = slip(11.067463450361,0.799344802313826)

    0.71+071 = 1*71*(101/100)
    1.42+142 = 2*71*(101/100)

  95. Paul Vaughan says:

    hhear dog Un?

    19.8589050137632 = beat(29.4571309198874,11.861990807677)
    6.57038184300286 = axial(14.7285654599437,11.861990807677)
    4.7320832829358 = axial(16.9132450828034,6.57038184300286)
    100.985205171956 = slip(19.8589050137632,4.7320832829358)
    1186.18276371796 = slip(100.985205171956,19.8589050137632) — ADJ0! (“well…B a dog”)
    3455.60863545854 = slip(100.985205171956,16.9132450828034) ~= 3456
    1727.80431772927 = slip(100.985205171956,8.4566225414017)

  96. Paul Vaughan says:

    Some (naively enough) believe expression reflects belief.

    casually exploring (just a little bit at a time) where some of the narratives came from

    Jupiter-Neptune Standish (1992) sidereal

    12.7821002221242 = beat(164.786005834669,11.861990807677)
    6.39105011106212 = 12.7821002221242 / 2
    3.19552505553106 = 12.7821002221242 / 4
    22.1309057968232 = harmean(164.786005834669,11.861990807677)
    11.0654528984116 = axial(164.786005834669,11.861990807677)
    5.53272644920579 = 11.0654528984116 / 2
    2.7663632246029 = 11.0654528984116 / 4

    22.2270434142659 = beat(5116.67146563023,22.1309057968232)
    11.113521707133 = 22.2270434142659 / 2
    5.55676085356648 = 22.2270434142659 / 4
    44.0711924742534 = harmean(5116.67146563023,22.1309057968232)
    22.0355962371267 = axial(5116.67146563023,22.1309057968232)
    11.0177981185634 = 22.0355962371267 / 2
    5.50889905928168 = 22.0355962371267 / 4

    85.1348300394896 = slip(22.2270434142659,12.7821002221242)
    85.1348300394896 = slip(12.7821002221242,11.113521707133)
    42.5674150197448 = slip(12.7821002221242,5.55676085356648)

    79.8222705138594 = slip(22.0355962371267,12.7821002221242)
    79.8222705138594 = slip(12.7821002221242,11.0177981185634)
    39.9111352569297 = slip(12.7821002221242,5.50889905928168)

    note well:
    555.982859223482 = slip(85.1348300394896,22.1309057968232)
    556 = 378 + 178

    378.378118042886 = slip(85.1348300394896,2.7663632246029)
    378.378 = 0.378+378 = 378*(1001/1000)

    326.007703249488 = slip(79.8222705138594,12.7821002221242)
    163.003851624744 = slip(79.8222705138594,6.39105011106212)

  97. Paul Vaughan says:

    supplementary
    1524.87927002403 = slip(164.786005834669,11.861990807677)
    5116.67146563023 = slip(1524.87927002403,12.7821002221242)
    1524.87927002403 = 1/(-1/11.861990807677+14/164.786005834669)
    5116.67146560344 = 1/(120/11.861990807677-1667/164.786005834669)

    378.378118034515 = 1/(-7444/11.861990807677+103412/164.786005834669)

    326.007703249529 = 1/(-1439/11.861990807677+19991/164.786005834669)
    163.003851624764 = 1/(-2878/11.861990807677+39982/164.786005834669)

    review
    200 = 378-178
    323 = 196883-196560
    0 = 1^2+2^2+3^2+4^2+5^2-28^2+27^2
    55 = 1^2+2^2+3^2+4^2+5^2 = 28^2-27^2 = 378-196883+196560
    110 = 1^2+2^2+3^2+4^2+5^2+28^2-27^2

    220 = s(s(220))

    178 = 2+3+5+7+11+13+17+19+23+31+47
    378 = 2+3+5+7+11+13+17+19+23+29+31+41+47+59+71

    556 = 178+378
    178+378*3-(196883-196560)*2 = 178+378+(28^2-27^2)+(5^2+4^2+3^2+2^2+1^2)

    recall
    70^2 = 24^2+23^2+22^2+…+5^2+4^2+3^2+2^2+1^2

    ‘n’so (apparently) an attempt to “contain” was MAID
    maybe some would be persuaded less by objective (and uninteresting) numerical facts than by superstition (about prescribed nicknames — e.g. “Beast”, “Monster”, “Amicable”)

    378±55
    ±
    178±55

    Only Standish (1992) sidereal ties in “the superstitious imagination”.

  98. Paul Vaughan says:

    EV
    1.59868960462765 = beat(1.0000174152119,0.615197263077614)
    0.761766203759125 = harmean(1.0000174152119,0.615197263077614)
    SEV
    0.781988555171283 = beat(29.4571309198874,0.761766203759125)
    36.0146108724673 = slip(1.59868960462765,0.781988555171283)
    36.014610872468 = 1/(-1.5/0.615197263077614+2.5/1.0000174152119-1/29.4571309198874)
    UEV
    0.768736359939692 = beat(84.01495797691,0.761766203759125) ; / 2 = 0.384368179969846
    10.0378419482516 = slip(1.59868960462765,0.384368179969846)
    36.0045726235707 = slip(10.0378419482516,1.59868960462765)
    36.0045726235671 = 1/(19/0.615197263077614-31/1.0000174152119+12/84.01495797691)
    NEV
    0.765304020794554 = beat(164.786005834669,0.761766203759125)
    17.9708503715071 = slip(1.59868960462765,0.765304020794554); * 2 = 35.9417007430143
    35.9417007430146 = 2/(-1.5/0.615197263077614+2.5/1.0000174152119-1/164.786005834669)
    adjustin’ wonder Standish (1992)
    sidereal in some hindsight to 163

  99. Paul Vaughan says:

    quick note on anomalistic “110.5

    1.59867106414771 = beat(1.0000262476142,0.615197860179071) ; / 4 = 0.399667766036927
    0.761769224080824 = harmean(1.0000262476142,0.615197860179071)
    0.715803548953639 = axial(11.8627021700857,0.761769224080824) ; / 4 = 0.17895088723841
    24.0670906604157 = slip(1.59867106414771,0.17895088723841)
    442.12002147416 = slip(24.0670906604157,1.59867106414771)
    110.53000536854 = slip(24.0670906604157,0.399667766036927)
    110.53000536789 = 1/(-416/0.615197860179071+656/1.0000262476142+240/11.8627021700857)

  100. Paul Vaughan says:

    interpretive note
    6.84967828238651 = slip(1.59867106414771,0.715803548953639)
    24.0670906604158 = slip(6.84967828238651,1.59867106414771)
    11.8627021700857 = harmean(44.2411450188424,6.84967828238651)
    also note
    23.0043337056467 = slip(11.0602862547106,0.761769224080824)

  101. Paul Vaughan says:

    “wwould
    $11 go Doww?
    Non! ‘U in?’ N/A the 8*R”U 0˚TAno.morsetllin’US
    19.8630730877524 = beat(29.4511026866654,11.862499899747)
    16.912768715208 = harmean(29.4511026866654,11.862499899747)
    6.56993811757712 = axial( 29.4511026866654 / 2 , 11.862499899747 )
    4.73181582715489 = axial(16.912768715208,6.56993811757712) ; / 2 = 2.36590791357745
    $isle llowe
    sci. in$$wwhen?
    50.2176861731761 = slip(19.8630730877524,2.36590791357745)
    890.599481124359 = slip(50.2176861731761,9.93153654387618)
    1983.88441752422 = slip(890.599481124359,65.7216461925253) ~= 1984 -well + N/A T = 0˚ NY
    “(!w)SST her(dup)(ll)c(!)[teach0r(45)]ty”
    1983.88441752493 = slip(815.681453466242,65.7216461925253) ~= 1984
    815.681453466242 = slip(100.435372346352,8.456384357604)
    100.435372346352 = slip(19.8630730877524,4.73181582715489)

  102. Paul Vaughan says:

    “The X ai br(!)” (tie mmminD-UK$won10)
    6.56993811757712 = axial( 29.4511026866654 / 2 , 11.862499899747 )
    65.7216461925253 = slip(30.5062369251788,19.8630730877524)
    61.0124738503575 = slip(29.4511026866654,11.862499899747) ; / 2 = 30.5062369251788
    851.495746676794 = slip(61.0124738503575,19.8630730877524) ——- point
    851.495746676803 = slip(19.8630730877524,6.56993811757712) ; / 2 = 425.747873338402
    851.495746676791 = harmean(890.599481124359,815.681453466242) —- of clarification
    9696.53769497082 = beat(890.599481124359,815.681453466242)
    / 96 + 378 + 178 ; / 10 ; / 10
    6464.35846333019 = slip(851.495746676803,19.8630730877524)
    6464.35846332961 = slip(980.671364041803,851.495746676794)
    / 64 + 378 + 178 ; / 10 ; / 10
    980.671364041803 = slip(425.747873338401,19.8630730877524)
    1799.77042668346 = slip(851.495746676803,4.96576827193809)

  103. Paul Vaughan says:

    576.095410986295 = slip(131.443292385051,16.912768715208)
    576.095410986304 = slip(65.7216461925253,16.912768715208)
    576 = 24^2
    1504.77950448739 = slip(576.095410986304,131.443292385051)
    851.495746676802 = harmean(980.671364041803,752.389752243925)
    6464.35846333058 = slip(851.495746676794,752.389752243925)
    “sh!tar$ PR10 in T = 00˚Knight$ guy”

  104. Paul Vaughan says:

    PRimITve(mm)y˚ThiCOrreryUN˚Tru()pll$

    1 = X1^2
    1 = X1 ———————– err pla in$
    3 = 2*X1+1
    4 = 2*X1+2*X1^2
    5 = 2*X1+2*X1^2+1
    9 = 3^2
    16 = 4^2
    25 = 5^2
    0 = 25-16-9
    145 = 3^4+4^3
    29 = (3^4+4^3)/5

    “Other examples of power mean numbers n such that some power mean of the divisors of n is an integer are the RMS numbers A140480.”

    “RMS numbers: numbers n such that root mean square of divisors of n is an integer.”
    “1, 7, 41, 239, 287, 1673, ”
    1, 5, 29, 169, 145, 845

  105. oldmanK says:

    Back to the old nut.
    See https://thatsmaths.com/2019/12/12/the-intermediate-axis-theorem/
    According to this, Iz the axial pole, is greater than Ix, Iy (Ix=Iy). But when can Ix (=the axis in line with the planets conjunction) due to geoid distortion, be greater than Iz, where then Iy<Iz<Ix. That is, Iz becomes the intermediate axis.

  106. Paul Vaughan says:

    “The Earth [weather whole$sum or piecewhy$???] is rotating about the axis with largest moment of inertia and smallest kinetic energy, so no catastrophic overturn is likely any time soon.” -DO!ugh!

  107. Paul Vaughan says:

    “CO
    up D
    R eelly”

    4 = X2^2
    2 = X2
    5 = 2*X2+1
    12 = 2*X2+2*X2^2
    13 = 2*X2+2*X2^2+1
    25 = 5^2
    144 = 12^2
    169 = 13^2
    0 = 169-144-25

    9 = X3^2
    3 = X3
    7 = 2*X3+1
    24 = 2*X3+2*X3^2
    25 = 2*X3+2*X3^2+1
    49 = 7^2
    576 = 24^2
    625 = 25^2
    0 = 625-576-49

    432 = 576-144

    25 = X5^2
    5 = X5
    11 = 2*X5+1
    60 = 2*X5+2*X5^2
    61 = 2*X5+2*X5^2+1
    121 = 11^2
    3600 = 60^2
    3721 = 61^2
    0 = 3721-3600-121

    36 = X6^2
    6 = X6
    13 = 2*X6+1
    84 = 2*X6+2*X6^2
    85 = 2*X6+2*X6^2+1
    169 = 13^2
    7056 = 84^2
    7225 = 85^2
    0 = 7225-7056-169

    3456 = 7056-3600

    “USA(mm)[R](!)sh(!)
    (mm)hhi(!)T˚(mou)”

    216 = average(-(12^2),(24^2))

  108. Paul Vaughan says:

    hey R(!)ch IT $not: weather˚Table 3456/432
    tsibxxrsolleurprocseapayyuanat(1)0sh(Un)
    $$˚Tratemi$$dupe ll!c ITe^he

    576 = X24^2
    24 = X24
    49 = 2*X24+1
    1200 = 2*X24+2*X24^2
    1201 = 2*X24+2*X24^2+1
    2401 = 49^2
    1440000 = 1200^2
    1442401 = 1201^2
    0 = 1442401-1440000-2401

    Without long-lasting, good (& willing) leaders,
    short-lasting bad puppet$(WHO Alliez don˚russT)R-sstrungllthruD-vll0id.

    625 = X25^2
    25 = X25
    51 = 2*X25+1
    1300 = 2*X25+2*X25^2
    1301 = 2*X25+2*X25^2+1
    2601 = 51^2
    1690000 = 1300^2
    1692601 = 1301^2
    0 = 1692601-1690000-2601

    200 = 2601-2401 = 378 – 178
    C ll0˚C nos… = isle owe sci. B ein

  109. Paul Vaughan says:

    soll(!)’ZZdeep(!p)V≡5(!MM)T˚hhP0RP(5!30)

    IT$hh&llIERS$T[˚]A(!)Un$ (b0.b0b) ~=
    909.091872159363 = harmean(936.563824740778,883.185646071486)

    Don˚T russhh(c)yaWWeather(W)P0RP(W)

  110. Paul Vaughan says:

    anomalistic ITCZwrite tooknow the poor P.S.

    Ruse “a rich(!)
    white now”
    make in pla(!)n
    COg node of moon0Rwwwllf a(!)r
    0.0740045002653102 = axial(11.0602862547106,0.0745030006844627)
    27.0301437219046 = 365.25 * 0.0740045002653102
    ˚Knew GBhhouRuse well make’in fine’weather feign moon$myth˚Trus˚T
    c(ll)a(!$)f(d)h(!) “a w(!)sh w rite no. w” hale?(!ll!m)a($www)ITCZ(mmm)rite[2˚K]now
    (od)r(mm.0u$nd)weatherIT$a(!)TSI($un11)

  111. Paul Vaughan says:

    4724
    =
    70^2+ΣΦ(323)

    24^2+Σφ(323)

  112. Paul Vaughan says:

    nodal confounding with stackplot convention
    0.0740050186809015 = axial(11.0718779353626,0.0745030006844627)
    0.0739217800064733 = axial(65.7216461925254,0.0740050186809015)
    27 ~= 26.9999301473644 = 365.25 * 0.0739217800064733

  113. Paul Vaughan says:

    5256
    =
    70^2+ΣΦ(323)

    44+Σφ(323)

    5256
    =
    70^2+ΣΦ(323)

    (σ^2)(22)

  114. Paul Vaughan says:

    5256 = 73 * 72

    2320 = (4*73)^2 – (4*72)^2
    2320 = (19+43+67+163)^2 – (24^2/2)^2

    21316 = 4*73^2

    292 = 4*73
    21315 = 4*73^2-1
    21317 = 4*73^2+1

    85264 = 292^2
    454329225 = 21315^2
    454414489 = 21317^2

    0 = 454414489-454329225-85264

    —————————————-

    20736 = 4*72^2

    288 = 4*72
    20735 = 4*72^2-1
    20737 = 4*72^2+1

    82944 = 288^2
    429940225 = 20735^2
    430023169 = 20737^2

    0 = 430023169-429940225-82944

  115. Paul Vaughan says:

    adjacent PPTs
    image bottom ray

    2320 = average(1184,3456)
    7920 = 2*(3456+504) ; 504 = 220+284 ; s(284) = 220
    5256 = 7920-2400-240-24
    tip of iceberg
    3456 = 126150-122694
    1728 = average(126150,-122694)

    —————————————-

    20164 = 4*71^2

    284 = 4*71
    20163 = 4*71^2-1
    20165 = 4*71^2+1

    80656 = 284^2
    406546569 = 20163^2
    406627225 = 20165^2

    0 = 406627225-406546569-80656

    —————————————-

    right-triangle
    2863146 = 284*20163 / 2
    2985840 = 288*20735 / 2
    3111990 = 292*21315 / 2
    area = base*height / 2
    122694 = 2985840-2863146
    126150 = 3111990-2985840
    double-difference ties in amicably

  116. Paul Vaughan says:

    consider: least-squares estimation, assumptions (measurement methods included)
    near-isosceles? special case of previously noted generalizations
    worthwhile to slowly mix review into pythagorean context
    as before: selective presentation only, as noteworthy details are grossly voluminous

  117. Paul Vaughan says:

    323 = 196883-196560 ; σ(323) = average(-163,883)
    4724 = 70^2+ΣΦ(323)–24^2-Σφ(323) = σ(4723)

  118. Paul Vaughan says:

    2401 = 49^2 = 7^2^2
    4724 = 70^2-24^2+s(7^2^2)
    2362 = average(-(24^2),70^2)+(378-178)
    4670 = 99^2-70^2-24^2+345

  119. Paul Vaughan says:

    test comment
    11.0602862547106
    22.1205725094212
    44.2411450188424

  120. Paul Vaughan says:

    test comment (continued)
    3 20 119 696
    4 21 120 697
    5 29 169 985
    9 400 14161 484416
    16 441 14400 485809
    25 841 28561 970225
    Pell PPT 7=3+4; 41=20+21; 239=119+120
    RMS nos. 1, 7, 41, 239, 287=7*41, 1673=7*239, 9799=41*239, 68593=7*41*239
    12 = 3+4+5 = perimeter
    70 = 20+21+29 ; 58 = 70-12 adjacent perimeter difference
    408 = 119+120+169 ; 490 = 12+70+408
    210 = 20*21 / 2 = 7# primorial
    140 = 210-70 = 169-29 ; h(140) = 5 = h(496) ; 140 = harmonic divisor no.
    99 = 119-20 = 120-21
    99^2 = 9801

    9799: 1,41,239,9799
    4901 = √average(1^2,41^2,239^2,9799^2)
    4900 = 70^2
    7 = √(average(-1,√9801)) ; m = 7^2
    504 = 220+s(220)
    s(2401) = ΣΦ(323)+Σφ(323) = 504-104

    review
    378 = average(average(220,284),average(220+284))
    178 = average(average(220,284),average(220+284)) – average(-104,220+284)
    200 = average(-104,220+284)
    400 = 504 – 104
    104 = 2*(378+178-504)

  121. Paul Vaughan says:

    also 577 = 697-120 = 985-169 = 400+177 ; 177 = 47+59+71 (top 3 M)

    1 1
    2 4
    3 9
    4 16
    5 25
    6 36
    7 49
    8 64
    9 81
    10 100
    11 121
    12 144
    13 169
    14 196
    15 225
    16 256
    17 289
    18 324
    19 361
    20 400
    21 441
    22 484
    23 529
    24 576

    select (contiguous) square sums
    1…7: 140

    2…4: 29 = 145 / 5 = (3^4+4^3) / 5
    8…9: 145 = 3^4+4^3
    8…10: 245 = half perimeter of 1st 3 Pell PPTs = 490/2 = (12+70+408)/2

    amicable
    2…9: 284 = s(220)
    5…15: 1210 = s(1184) ; 400 = 5^2^2 – 15^2

    M
    7…9: 194
    7…10: 294 = σ(194)

    casual, meandering scenic route

    with sociable review:

    Σδ(378) = 752 = 1504 / 2
    Σδ(752) = 894
    Σδ(894) = 902
    Σδ(902) = 894
    Σδ(894) = 902
    Σδ(902) = 894 = 2 * 447 (405ka near-isosceles)

  122. Paul Vaughan says:

    clarification 553
    4724 = 4901-177
    (attest comment (was) caught in the filter)

  123. Paul Vaughan says:

    4724 = 99^2-70^2-71-59-47
    2362 = average(99^2,-(70^2+47+59+71))

    4724 = 2*(70^2-23^2-22^2-21^2-20^2-19^2-323)
    2362 = 70^2-23^2-22^2-21^2-(20^2+323)-19^2

  124. Paul Vaughan says:

    weather allies or all lies AB?
    2362 = 24^2+18^2+17^2+16^2+15^2+14^2+13^2+12^2+11^2+10^2+9^2+8^2+7^2+6^2+5^2+4^2+3^2+2^2+1^2-323
    2362 = 99^2-70^2-23^2-22^2-21^2-20^2-19^2-18^2
    2362 = 99^2-2*70^2+24^2+17^2+16^2+15^2+14^2+13^2+12^2+11^2+10^2+9^2+8^2+7^2+6^2+5^2+4^2+3^2+2^2+1^2

    323 = 196883196560
    s(323)=37, 684, 685 ; ^2: 1369+467856=469225

  125. Paul Vaughan says:

    Such clean “answers” actually raise questions about measurement and estimation methods. The “leadership” failed catastrophically. Trump: OUT! NOW!

  126. Paul Vaughan says:

    herdiamondave “BR˚™I Pence sol”
    7 ; 7^2 = 49
    2*7+1 ; 2*(7+49) ; 2*56+1 ;; ^2
    15 112 113 ; 240 = 15+112+113
    225 12544 12769 ; 225 = 12769-12544 = 112+113
    240 = perimeter
    σ(240) = 744
    s(240) = 504 = 220+284
    s(225) = 178
    σ(225) = 403

  127. Paul Vaughan says:

    Bye Don

    75 5625 151 11400 11401 0= -22801 -129960000 +129982801
    76 5776 153 11704 11705 0= -23409 -136983616 +137007025

    76 D˚Tales
    76 ; 76^2 = 5776
    76*2+1 ; 2*(76+76^2) ; 2*(76+76^2)+1
    153 11704 11705 ; ^2 =
    23409 136983616 137007025

    change from previous triangle in series:
    304 304 0 608

    153 = sum of non163 Heegner nos. = 316-163
    s(304) = 316 = sum of Heegner nos. (incl. 163)
    s(608) = 652 ; s(652) = 496 ; s(496) = 496

    time 304
    “WHO ami 2 D[˚˚]agree?
    T˚Ravell the www urll
    DandD the 7˚˚CCCCCCC
    EVorrery buddy’
    s(ll˚˚K)in 4*something” 316 index

    76 = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24 = 19 + 19 + 19 + 19

  128. Paul Vaughan says:

    at least-squares imagination (not forthcoming by dawn)

    The obvious problem that quickly becomes eminently clear with PPTs is the sheer volume of simple sporadic group examples. We could (simply enough) be left giving (nearly countless) examples for decades.

  129. Paul Vaughan says:

    test comment (supplementary) 5pelltaxicab
    12 = 3+4+5 = perimeter
    70 = 20+21+29
    408 = 119+120+169 ; 490 = 12+70+408
    average perimeter of adjacent PPTs in series:
    41 = average(12,70)
    239 = average(70,408)
    9799 = average(12,70)*average(70,408) = 41*239

  130. Paul Vaughan says:

    Inclusive Translation (heard well by 194 message interpreters)

    generalization: “power mean numbers n such that some power mean of the divisors of n is an integer” (quote from OEIS A001599)

    feature of least-squares estimation:
    RMS = root mean square

    RMS nos. relations (e.g. 9799 from A140480) with
    Pell PPTs (along 45˚)
    and the other near-isosceles triangles with 0˚ & 90˚ limits (series relating to one another 1:1)


    50: 200 9999 10001
    9799 = 9999-200
    9801 = 10001-200

    corresponds with

    90˚
    49: 99 4900 4901
    50: 101 5100 5101
    9801 = 99^2
    200 = 5100-4900 = 5101-4901

    exploring (hierarchically-nested) integer scale, first note
    23: 47 1104 1105
    24: 49 1200 1201
    25: 51 1300 1301
    200 = 51^2 – 49^2
    2450 = 49+1200+1201
    194 = 2450-(47+1104+1105)
    areas
    25: 33150 = 51*1300/2
    24: 29400 = 49*1200/2
    23: 25944 = 47*1104/2
    differences
    3456 = 29400-25944
    3750 = 33150-29400

    the double-difference is subtle:
    294 = 3750-3456 = σ(194)

  131. Paul Vaughan says:

    clarification

    “The RMS values (A141812) of prime RMS numbers [A140480] are prime Pell numbers (A000129) with an odd index.”

    almost-isosceles PPTs
    =
    If a, b, c are the sides of this type of primitive Pythagorean triple (PPT) then the solution to the Pell equation is given by the recursive formula

    a(n) = 6*a(n−1) − a(n−2) + 2
    with a(1) = 3 & a(2) = 20

    b(n) = 6*b(n−1) − b(n−2) − 2
    with b(1) = 4 & b(2) = 21

    c(n) = 6*c(n−1) − c(n−2)
    with c(1) = 5 & c(2) = 29
    =

    n: a(n) b(n) c(n) perimeter=a(n)+b(n)+c(n)
    1: 3 4 5 12
    2: 20 21 29 70
    3: 119 120 169 408
    4: 696 697 985 2378

  132. Paul Vaughan says:

    simpler Pell PPT generation aware of perimeter

    first generate A000129:
    “Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).”
    “0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860”

    then:
    (higher odd-position index) – (lower even-position index)
    = perimeter – hypotenuse = NSW nos.
    7 = 12 – 5
    41 = 70 – 29
    239 = 408 – 169
    1393 = 2378 – 985
    8119 = 13860 – 5741

    finally, split:
    floor
    3 = ⌊average(-5,12)⌋ = ⌊7/2⌋
    20 = ⌊average(-29,70)⌋ = ⌊41/2⌋
    119 = ⌊average(-169,408)⌋ = ⌊239/2⌋
    696 = ⌊average(-985,2378)⌋ = ⌊1393/2⌋
    4059 = ⌊average(-5741,13860)⌋ = ⌊8119/2⌋
    +ceiling
    4 = ⌈average(-5,12)⌉ = ⌈7/2⌉
    21 = ⌈average(-29,70)⌉ = ⌈41/2⌉
    120 = ⌈average(-169,408)⌉ = ⌈239/2⌉
    697 = ⌈average(-985,2378)⌉ = ⌈1393/2⌉
    4060 = ⌈average(-5741,13860)⌉ = ⌈8119/2⌉
    such that
    3+4+5 = 12
    20+21+29 = 70
    119+120+169 = 408
    696+697+985 = 2378
    4059+4060+5741 = 13860

  133. Paul Vaughan says:

    near-isosceles PPT generation (concisely)

    1: vertical (90˚)
    a(n) = 2*n + 1 ; b(n) = 2*(n+n^2) ; c(n) = b(n) + 1
    e.g.
    1: 3 4 5
    2: 5 12 13
    3: 7 24 25
    4: 9 40 41
    5: 11 60 61

    2: horizontal (0˚)
    a(n) = 4*n ; b(n) = 4*(n^2) 1 ; c(n) = 4*(n^2) + 1
    e.g.
    1: 4 3 5
    2: 8 15 17
    3: 12 35 37
    4: 16 63 65

    Pell (45˚)

  134. Paul Vaughan says:

    Recommendation: Set up tables (for example in Excel) for the 3 near-isosceles PPT series. For each row calculate simple things like perimeter & area. Then difference (and double-difference) things for adjacent PPTs (rows).

    Having given a concise foundation (see last comment) I’m going to start simply copying/pasting whole rows from such a table into comments (without explanation, which is elementary enough).

    Integer-scaling patterns in the tables match simplified (rounded-off) narratives.
    There isn’t time to do more than rough out a selection of dots (no exhaustive treatment) for others to connect (or even more efficiently: choose not to connect).

  135. Paul Vaughan says:

    15 & 1800

    2 (horizontal) :
    15 900 60 899 901 0 3600 808201 811801 116 4 116 116 0 464 195112 195576 1860 236 16 26970 5046 672 48 839 841 804601 15

    1 (vertical) :
    225 50625 451 101700 101701 0 203401 10342890000 10343093401 449 2 900 900 0 1800 182250000 182251800 203852 1802 8 22933350 303750 2694 12 101249 101250 10342686599 225

  136. Paul Vaughan says:

    405….k

    σ(223) = 224
    σ(224) = 504

    224 50176 449 100800 100801 0 201601 10160640000 10160841601 447 2 896 896 0 1792 179830784 179832576 202050 1794 8 22629600 301056 2682 12 100351 100352 10160438399

    1800 = 201601-203401

    “you’ve got THE BLUES too
    all that Ny^e˚T˚nd all˚Th’Un’X˚T
    (M s)a(m w)yth outlookkingback
    MAID(h)˚Th’www’$$T˚earn($lloodd)
    (hll!$)$(!)lly˚F($1100)” hear˚T[he][earth] BR(!)CO(np!dn)h!

  137. Paul Vaughan says:

    224 200704 896 200703 200705 0 802816 40281694209 40282497025 1788 4 1788 1788

    225 202500 900 202499 202501 0 810000 41005845001 41006655001 1796 4 1796 1796

    1796 = 896+900 = 202499-200703 = 202501-200705 = 894+902
    σ(894) = 1800

    Σδ(378) = 752 ; Σδ(752) = 894
    Σδ(894) = 902 ; Σδ(902) = 894

    224 50176 449 100800 100801 0 201601 10160640000 10160841601 447 2 896 896 0 1792 179830784 179832576 202050 1794 8 22629600 301056 2682 12 100351 100352 10160438399 897 224 896 200704 200706 200703 100353 0 898

    225 50625 451 101700 101701 0 203401 10342890000 10343093401 449 2 900 900 0 1800 182250000 182251800 203852 1802 8 22933350 303750 2694 12 101249 101250 10342686599 901 225 900 202500 202502 202499 101251 0 902

  138. Paul Vaughan says:

    234 = 2*117 = 163+71 ; 117 = 58+59
    adjacent perimeters m = 233, 234
    937 = average(-467-109044-109045,469+109980+109981)

  139. Paul Vaughan says:

    1106 = 7*158
    2212 = 14*158
    4424 = 28*158 ; 28 = s(28)

  140. Paul Vaughan says:

    3-4-5 & 20-21-29 : 15 & 290

    41 = 21^2 – 20^2 = 441 – 400

    40 1600 81 3280 3281 0 6561 10758400 10764961 79 2 160 160 0 320 1024000 1024320 6642 322 8 132840 9600 474 12 3199 3200 10751839 161 40 160 6400 6402 6399 3201 0 162

    41 1681 83 3444 3445 0 6889 11861136 11868025 81 2 164 164 0 328 1102736 1103064 6972 330 8 142926 10086 486 12 3361 3362 11854247 165 41 164 6724 6726 6723 3363 0 166

    163 26569 327 53464 53465 0 106929 2858399296 2858506225 325 2 652 652 0 1304 69291952 69293256 107256 1306 8 8741364 159414 1950 12 53137 53138 2858292367 653 163 652 106276 106278 106275 53139 0 654

    164 26896 329 54120 54121 0 108241 2928974400 2929082641 327 2 656 656 0 1312 70575104 70576416 108570 1314 8 8902740 161376 1962 12 53791 53792 2928866159 657 164 656

  141. Paul Vaughan says:

    the obvious twin prime pair in PPT series 2: 3, 5
    15 = 3*5

    10 = 13-3
    13 = 117-104
    18 = 13+5
    22 = 18+4

    553 = 657 – 104 ; 1106 = 2*553 ; 2212 = 2*1106 ; 4424 = 2*2212

    11 = average(18,4)
    11^2 = 121
    63 = 10+13+18+22
    58 = 121-63
    19 = x mod 24 for x = 19, 43, 67, 163
    158 = average(1+2+3+7+11, 19+43+67+163)
    37 = 158-121 = s(323) = s(17*19) = 17+19+1 ; σ(323) = 360

    σ(290) = 540 ; 657 =117+540
    σ(145) = 180 ; 297 = 117+180
    194 = Σs(145)

    400 = 20^2 = s(7^2^2) = ΣΦ(323)-Σφ(323) = 447-47 = 993-593 ; 993 = (σ^2)(400)
    323 = 196883-196560

    296 87616 593 175824 175825 0 351649 30914078976 30914430625 591 2 1184 1184 0 2368 414949376 414951744 352242 2370 8 52131816 525696 3546 12 175231 175232 30913727327 1185 296 1184 350464 350466 350463 175233 0 1186

    297 88209 595 177012 177013 0 354025 31333248144 31333602169 593 2 1188 1188 0 2376 419169168 419171544 354620 2378 8 52661070 529254 3558 12 176417 176418 31332894119 1189 1485 900 1485 297 1188 352836 352838 352835 176419 0 1190

    145 = 3^4 + 4^3 ; 29 = (3^4 + 4^3)/5
    290 divides 58

    158 = 10+13+18+22+37+58
    239 = average(158,320)
    320 = 28+163+67+43+19

  142. Paul Vaughan says:

    3*5 = 15
    15^2 = 225 = φ(657)

    1: 1
    2: 1 2
    3: 1 3
    12: 1 2 3 4 6 12
    17: 1 17
    28: 1 2 4 7 14 28
    32: 1 2 4 8 16 32
    72: 1 2 3 4 6 8 9 12 18 24 36 72
    108: 1 2 3 4 6 9 12 18 27 36 54 108
    117: 1 3 9 13 39 117
    297: 1 3 9 11 27 33 99 297
    657: 1 3 9 73 219 657

    unique composite sum
    432 = 4+6+8+12+14+16+18+28+24+32+36+54+72+108 = 2 * 216 = Φ(657)

  143. Paul Vaughan says:

    miscellaneous notes

    290+15^2 = Σs(178) = 657-171-71
    ΣΦ(216) = ΣΦ(117) = ΣΦ(252) = 111 ; 223 = ΣΦ(504)
    φ(290) = s(15^2) = s(225) = 178 ; δ(657) = 225

    Φ(1806) = 504
    504 = δ(744) = σ(220) = σ(284) = 400+104 = 657-153
    153 = 316-163 ; 158 = ΣΔ(153)

    δ(252) = 180 = Δ(297) = Φ(297) ; 297 = 117+180 ; Δ(894) = 297
    δ(504) = 360 = σ(323) ; 657 = 297+360

    735 = ΣΦ(1800) = Σφ(432) ; 432 = Δ(657)
    φ(837) = 297 ; φ(297) = 117 = Δ(354)

    Φ(378)+Φ(216) = 180 ; Φ(378) = 108 ; Φ(216) = 72
    378-216 = 162 = ΣΔ(378) = Φ(163) ; 163 = ΣΦ(378)

    28 = s(28) ; 6 = s(6)
    34 = 6 + 28 = 2 * 17

    17 289 35 612 613 0 1225 374544 375769 33 2 68 68 0 136 78608 78744 1260 138 8 10710 1734 198 12 577 578 373319 69 17 68 1156 1158 1155 579 0 70

    1225 1500625 2451 3003700 3003701 0 6007401 9.02221E+12 9.02222E+12 2449 2 4900 4900 0 9800 29412250000 29412259800 6009852 9802 8 3681034350 9003750 14694 12 3001249 3001250 9.02221E+12 4901 1225 4900 6002500 6002502 6002499 3001251 0 4902

  144. Paul Vaughan says:

    supplementary notes on 104-level ties to A100570 (& 744-levels)

    10 = 5+5 ; 10^2 = 100 = 158-58
    17 = average(s(6),s(28)) ; 117 = 41+76
    17: 1 17 ; sum divisor^2 = 1^2 + 17^2 = 290
    11^2 – 10^2 = 58-37 = 21
    11^2 = 158-37
    140 = 158-18
    145 = 158-13 = average(1^2,17^2)

    24 = 37-13
    19 = 37-18
    19 = 163 mod 24 = 67 mod 24 = 43 mod 24 = 19 mod 24
    76 = 163 mod 24 + 67 mod 24 + 43 mod 24 + 19 mod 24

    28 = 10+18
    32 = 10+22

    35 = 13+22

    47 = 10+37
    59 = 22+37
    71 = 13+58

    180 = 22+158
    216 = 58+158
    316 = 158+158

    168 = 10+158
    240 = 168 + 70 + (5-3)

    234 = 13*18
    1369 = 37^2

    2378 = 41*58
    902 = 41*37

    1 = 73 mod 24 ; next prime is 79
    158 = 31 + 31st prime = 2 * 79

    harmonic divisor nos.
    7 = 1+6 ; 7^2 = 49
    35 = 1+6+28 ; 35^2 = 1225

    1:	1																																															
    6:	1	2	3	6																																												
    28:	1	2	4	7	14	28																																										
    140:	1	2	4	5	7	10	14	20	28	35	70	140																																				
    496:	1	2	4	8	16	31	62	124	248	496																																						
    270:	1	2	3	5	6	9	10	15	18	27	30	45	54	90	135	270																																
    8128:	1	2	4	8	16	32	64	127	254	508	1016	2032	4064	8128																																		
    672:	1	2	3	4	6	7	8	12	14	16	21	24	28	32	42	48	56	84	96	112	168	224	336	672																								
    1638:	1	2	3	6	7	9	13	14	18	21	26	39	42	63	78	91	117	126	182	234	273	546	819	1638																								
    6200:	1	2	4	5	8	10	20	25	31	40	50	62	100	124	155	200	248	310	620	775	1240	1550	3100	6200																								
    2970:	1	2	3	5	6	9	10	11	15	18	22	27	30	33	45	54	55	66	90	99	110	135	165	198	270	297	330	495	594	990	1485	2970																
    8190:	1	2	3	5	6	7	9	10	13	14	15	18	21	26	30	35	39	42	45	63	65	70	78	90	91	105	117	126	130	182	195	210	234	273	315	390	455	546	585	630	819	910	1170	1365	1638	2730	4095	8190
    

    h(x) x
    1 1
    2 6
    3 28 = 4*(1+6)
    5 140 = 4*(1+6+28)
    5 496 = 270+15^2+1
    6 270 = 504 – 234
    7 8128; φ(8128) = 4096
    8 672 = 657+15
    9 1638
    10 6200
    11 2970 = 297*(5+5) = 297*(h(140)+h(496))
    15 8190

    432 = average(-12-32-72-108,17+117+297+657) without 1+2+3 = s(6) = 6 & 28 = s(28)
    657 = (1+2+3)+(28)+(17)+12+72+108+117+297
    657 = (2)+(32)+(17)+12+72+108+117+297 ; twin prime gap = 2 = 5-3 ; 2^5 = 32
    153 = 17 * 3^2 ; 13 = 9+5 ; 140 = 153-13 ; 37 = 2^5+5 ; 58 = 53+5

    prime+5 for 10, 18, 22, 58
    3^2 & 2^5 +5 for 13 & 37
    37-13 = 24 ; 12 = average(-13,37) ; 41 = 3^2+2^5 = 3^(5-3)+(5-3)^5

    9 81 19 180 181 0 361 32400 32761 17 2 36 36 0 72 11664 11736 380 74 8 1710 486 102 12 161 162 32039 37 9 36 324 326 323 163 0 38

  145. Paul Vaughan says:

    typo: 2^3 vs. 3^2
    2^3 & 2^5 +5 for 13 & 37

  146. Paul Vaughan says:

    tabulations

    products
    _	10	13	18	22	37	58	158
    10	100						
    13	130	169					
    18	180	234	324				
    22	220	286	396	484			
    37	370	481	666	814	1369		
    58	580	754	1044	1276	2146	3364	
    158	1580	2054	2844	3476	5846	9164	24964
    
    sums
    _	10	13	18	22	37	58	158
    10	20						
    13	23	26					
    18	28	31	36				
    22	32	35	40	44			
    37	47	50	55	59	74		
    58	68	71	76	80	95	116	
    158	168	171	176	180	195	216	316
    
    differences
    _	10	13	18	22	37	58	158
    10							
    13	3						
    18	8	5					
    22	12	9	4				
    37	27	24	19	15			
    58	48	45	40	36	21		
    158	148	145	140	136	121	100	
    

    3,4,5 PPT contains twin prime pair 3,5 ; 3*5 = 15
    difference diagonal sequence begins 3, 5, 4, 15 = 37-22
    145 = ( 3^4 + 4^3 ) / 5

    1 unknown remains.
    acceptable menu highlights (browse 58 rows & columns) :
    71 — M
    216 — Plato
    316 (also note 19, 24, & 76) — Heegner

    A020495 includes not 37, 22, 18, & 13 but 58 & 10 (lowest)

    _
    104-level review
    104.212132286568 = ⌊(e^√10π)^(1/2)⌉^2 – e^√10π
    -103.947369666712 = ⌊(e^√13π)^(1/2)⌉^2 – e^√13π
    104.007114381762 = ⌊(e^√18π)^(1/2)⌉^2 – e^√18π
    104.001742574386 = ⌊(e^√22π)^(1/2)⌉^2 – e^√22π
    -103.999977946281 = ⌊(e^√37π)^(1/2)⌉^2 – e^√37π
    104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π

  147. Paul Vaughan says:

    correction, recipe

    fact check: “1 unknown remains.”

    ((blueCOllAR˚Curious George aside inEUropeDon˚T!ache3,5twin prime debate 2^far with 2^3 = 8 = 18-10 & 2^5 = 32 = 22+10 no. win just in duh AR? earn true dough’s ENSO AR sh!op temp˚T=ice+yawn))

    start with lowest no. that’s neither square nor square + prime:
    10
    use pieces of 3,4,5 twin prime PPT to complete series
    13 = 10+3
    18 = 13+5
    22 = 18+4
    37 = 22+3*5
    58 = (10+13+18+22+37)-((3^4+4^3)/5+13) = 10+18+22+37-(3^4+4^3)/5
    158 = 10+13+18+22+37+58

  148. Paul Vaughan says:

    alternatively
    58 = ((3^4+4^3)+13)-(10+13+18+22+37) = (3^4+4^3)-(10+18+22+37)

    clarification
    (3^4+4^3) = 145 &
    (3^4+4^3)/5 = 29
    property of type 1 aiPPTs, including 3,4,5 which is shared with both Pell & type 2

    ((in blueCOllARease: 5un porpoise cc rev verse in clues of 11y))

  149. Paul Vaughan says:

    12 0˚Clock:
    functional numeracy debate
    why phi?sh!un+11=SST “c˚nUKsh!yuan is made”

    twin prime pair(s(n))+un=n˚yABlew
    water JSUNheard earn deep dive
    sales just in duh!enso!sh!op
    “CCpickup that gui˚Tar&talk 2^ME” P
    58 = average(-(3^4+4^3)/5,(3^4+4^3)) = average(-29,145)
    how too /5pell that with functional numeracy?
    58 = average(-√(20^2+21^2),145)
    58 = (20+21+29)-(3+4+5) = 70-12
    41 = 20+21 = 21^2-20^2
    type 1 PPT m = 41 hypotenuse length change from previous in series:
    164 = 3445-3281
    type 1 PPT m = 164 perimeter change from previous (m=163) in series:
    1314 = (329+54120+54121)-(327+53464+53465) ; ‘n’so:
    657 = average(-(327+53464+53465),(329+54120+54121))
    ˚TruMP (with “Far age”) : fool me once
    BoreUS (with drama queen) : fool me twice ??
    No. who‘s tri˚Cking now 2^further “justify” luckdown dumb*sses ITally?

    45˚Top write too bought’em left corner/5pell

  150. Paul Vaughan says:

    The Butterfly

    oeis is pretty useful (missing a lot, but not everything)
    https://oeis.org/search?q=4+1+3+2+1+4+1
    https://oeis.org/search?q=1+4+1+2+3+1+4 reverse

    too cue^BS
    note$well :

    96^3 = 884736

    96 = 24 * 4
    24 = 24 * 1
    24 = 8 * 3
    8 = 4 * 2
    4 = 4 * 1
    4 = 1 * 4
    1 = 1 * 1
    1 — start with 1 and build up

    don[˚T]yell’s myth 14
    selective note$ for 123
    an aztec poll lover 14

    163 = 67 + 96
    67 = 43 + 24
    43 = 19 + 24
    19 = 11 + 8
    11 = 7 + 4
    7 = 3 + 4
    3 = 2 + 1
    2 = 1 + 1
    1 — start with 1 (& bill dupe)

    m=37 (a 104-level)
    37^2 = 1369
    148 (hypotenuse change from previous in series)
    298 (perimeter change from previous) (remember that?)
    m=148: 593
    657-593 = 64
    https://oeis.org/A068227/b068227.txt
    64 4
    65 1
    66 3
    67 2
    68 1
    69 4
    70 1
    8^2 indexes multiplicity changes of Heegner no. differences (in reverse)
    look at rows m=35,36,37
    the butterfly, she see what he ignore:
    58 4
    59 1
    60 3
    61 2
    62 1
    63 4
    64 1
    https://oeis.org/A025428/b025428.txt
    discrete measure mean˚T f*IT square$ at least

    _
    miscellaneous notes

    3 4 5
    20 21 29
    differences
    17 17 24

    cumulative sum of Pell perimeters & double:
    12 24
    82 164
    490 980
    2868 5736
    16728 33456
    16-15=1

    837 116 845
    123 836 845
    PPT table
    116 = 29 * 4 = 2 * 58

  151. Paul Vaughan says:

    as a PR cedin’with blue collar ease:
    MEtri˚C moon sst˚ar hh0use pot 11

  152. Paul Vaughan says:

    review
    8 = 902-894
    64 = 284-220

    in excel, column A
    1 2 3 4 5 etc.
    enter following in cell B1, copy/paste downwards:
    =IF(A1<5,A1,IF(MOD(A1,5)=0,OFFSET($B$1,A1/5-1,0),OFFSET($B$1,A1-FLOOR(A1/5,1)-1,0)))
    23 1
    24 4
    25 1
    26 2
    27 3
    28 1
    29 4
    https://oeis.org/A255825

  153. Paul Vaughan says:

    A138967: “Infinite Fibonacci word on the alphabet {1,2,3,4}.”
    1
    4
    1
    2
    3
    1
    4

    1
    1 = 1 * 1
    4 = 4 * 1
    4 = 1 * 4
    8 = 2 * 4
    24 = 3 * 8
    24 = 1 * 24
    96 = 4 * 24

    1
    2 = 1 + 1
    3 = 1 + 2
    7 = 4 + 3
    11 = 4 + 7
    19 = 8 + 11
    43 = 24 + 19
    67 = 24 + 43
    163 = 96 + 67

    1 = (1+3) / 4
    2 = (1+7) / 4
    3 = (1+11) / 4
    5 = (1+19) / 4
    11 = (1+43) / 4
    17 = (1+67) / 4
    41 = (1+163) / 4

  154. Paul Vaughan says:

    Pell PPT 1 & 2
    3 4 5
    20 21 29

    1 = 4-3 = 21-20
    2 = 5-3 (twin prime PPT)
    3 = 3
    5 = 5
    11 = (29-20)+(5-3)
    17 = 20-3 = 21-4
    41 = 20+21 = 21^2-20^2
    2, 3, 5, 11, 17, 41

  155. Paul Vaughan says:

    lowest square = 0^2 = 0
    lowest semiprime = 2*2 = 4 ; therefore
    lowest square + semiprime = 0+4 = 4
    1, 2, & 3 can’t be expressed as square + semiprime
    cumulative sum of column 2 in column 3
    1 1 1
    2 2 3
    3 3 6
    4 12 18 18
    5 17 35 22 13
    6 28 63 22 18 13 10
    7 32 95 58 37

  156. Paul Vaughan says:

    “The terms of the sequence cannot be easily calculated, even with raw number crunching power. This keyword is most often used for sequences corresponding to unsolved problems […]” — “easy

    Φ(657) = 432 = 360+72 ; 72 = Φ(216)
    φ(657) = 225 = 117+108 ; 108 = Φ(378)

    σ(145) = 180 ; 194 = Σs(145)
    σ(323) = 360
    σ(290) = 540 ; (σ_2)(17) = 290
    φ(290) = s(225) = 178

    φ(297) = 117 ; Φ(117) = 72 = 180-108
    Φ(297) = Φ(836) = Φ(209) = 180 ; 297 = Δ(894)

    3,4,5 PPT contains
    divisor sum lowest
    twin prime product
    σ(15) = 24
    19 = x mod σ(3*5) for x = 163, 67, 43, 19
    247 = average(-163,657) = average(Σφ(323),ΣΦ(323)) = ΣΦ(490)

    7*19 = 133 ; Φ(133) = 108 = 216 / 2
    11*19 = 209 ; Φ(209) = 180 = 360 / 2
    13*19 = 247 ; Φ(247) = 216 = 158+58
    17*19 = 323 ; Φ(323) = 288 = 180+108
    323 = 196883-196560 ; σ(323) = 360
    ΣΦ(323) = 447 ; 47 = Σφ(323) ; average = 247

    A100570 mnemonic: 158-58 = 10+13+18+22+37 =
    100; 570 = 323+247

    odd prime divisor sum = 1+3+7+11+13+17+73 = 125
    6^3 = 3^3 + 4^3 + 5^3 ; 216 = 27 + 64 + 125

  157. Paul Vaughan says:

    64 = 657-593 = 284-220 = (902-894)^2

    171 = average(209,133)
    342 = 209+133
    76 = 209-133 = 163 mod σ(3*5) + 67 mod σ(3*5) + 43 mod σ(3*5) + 19 mod σ(3*5)

    342 = Σδ(220) = 2 *171
    142 = ΣΔ(220) = 2 * 71
    200 = 342-142 = 378-178

    400 = s(2401) = ΣΦ(323)-Σφ(323) = Σs(242)
    993 = 593+400 = (σ^2)(400) = 3 * 331
    902 = Σδ(894) = Σδ(Σδ(993)) = Σδ(Σδ(Σδ(378)))

    noteworthy midpoint
    64 64 311 4 = average(311 mod 4, 311 mod 6)
    65 65 313 1 = average(313 mod 4, 313 mod 6)
    66 66 317 3 = average(317 mod 4, 317 mod 6)
    67 67 331 2 = average(331 mod 4, 331 mod 6)
    68 68 337 1 = average(337 mod 4, 337 mod 6)
    69 69 347 4 = average(347 mod 4, 347 mod 6)
    70 70 349 1 = average(349 mod 4, 349 mod 6)

    “The name “genity” was derived from “genes” and “parity”, since the fourfold values of g(p) in a sequence corresponding to prime arguments resemble the genetic sequences of the nucleotides in the DNA.”

  158. Paul Vaughan says:

    RMS

    number: divisors
    1: 1
    7: 1 7
    41: 1 41
    239: 1 239
    287: 1 7 41 287
    1673: 1 7 239 1673

    √average(number divisor^2) = value ; e.g. 7:
    √average(1^2,7^2) = √average(1,49) = √25 = 5

    number value
    1 1
    7 5
    41 29 = (3^4 + 4^3)/5
    239 169
    287 145 = 3^4 + 4^3
    1673 845 = 5 * 169

    158 = 10+13+18+22+37+58 (104 levels)
    239 = average(158,320)
    320 = 28+163+67+43+19 (744 levels)

  159. Paul Vaughan says:

    starting with
    10 = 5 * 2
    58 = 29 * 2

    1+2+3+12+17+28+32 = 58+d
    1+2+3+12+17+28 = c+b+a+10
    1+2+3+12+17 = a+c
    1+2+3+12 = b

    “The terms of the sequence can be easily calculated.” — OEIS Wikipedia

    Curiously, the following sequence (of 104 levels) is linked to neither A020495 nor A100570 in the OEIS: 10, 13, 18, 22, 37, 58.

  160. Paul Vaughan says:

    A025428 “Number of partitions of n into 4 nonzero squares.”
    4
    58 = 7^2+2^2+2^2+1^2
    58 = 5^2+4^2+4^2+1^2
    58 = 5^2+5^2+2^2+2^2
    58 = 6^2+3^2+3^2+2^2
    1
    59 = 5^2+4^2+3^2+3^2
    3
    60 = 7^2+3^2+1^2+1^2
    60 = 5^2+5^2+3^2+1^2
    60 = 6^2+4^2+2^2+2^2
    2
    61 = 7^2+2^2+2^2+2^2
    61 = 5^2+4^2+4^2+2^2
    1
    62 = 6^2+4^2+3^2+1^2
    4
    63 = 6^2+5^2+1^2+1^2
    63 = 7^2+3^2+2^2+1^2
    63 = 5^2+5^2+3^2+2^2
    63 = 6^2+3^2+3^2+3^2
    1
    64 = 4^2+4^2+4^2+4^2

  161. Paul Vaughan says:

    Subtle Distinction

    4424 = 158*28 = 316*14 ——— 44.24114502 JEV
    2212 = 158*14 = 316*7 ———- 22.12057251 anomalistic˚Con’t ai’n’minT331
    1106 = 158*7 = 316*3.5 ——– 11.06028625 Standish (1992)

    553 = 657 – 104
    153 = 657 – 504 ; 504 = σ(220)
    400 = 553 – 153
    400 = 504 – 104
    163 = 316 – 153

    158 = 10+13+18+22+37+58 (104 levels)
    158 = 902 – 744 ; 744 = σ(240)
    245 = 902 – 657
    central being, so average
    237 = 894 – 657

    240 = 744 – 504
    76 = 316 – 240
    17 = 657 – 640 ; (σ_2)(17) = 290
    640 = 744 – 104
    320 = 28+163+67+43+19 (744 levels)
    64 = 284 – 220
    8 = 902 – 894

    mean˚Too laugh?doubt write go/5pell orrery:
    x ln(x)
    1 0
    2 0.693147181
    3 1.098612289
    7 1.945910149
    11 2.397895273
    19 2.944438979
    43 3.761200116
    67 4.204692619
    163 5.093750201
    sum:
    316 22.13964681
    22.13941145 —————— NASA ‘factsheet’ JEV sidereal
    measure weather number theory or knew miracll method

  162. Paul Vaughan says:

    nigh$awe˚CelleSST˚R(!)angll
    240 = σ(158) = σ(47+59+71)
    117 = ΣΦ(158) ; 158 = ΣΔ(153)
    234 = 2 * 117 = σ(153)
    2953 = 4724-47*59*71+⌊(47*59*71)^(1/3)⌉^3
    9.93288388379268 = (657+196883-196560)^(1/3)
    19.8657677675854 = (657+196883-196560)^(1/3) * 2
    7 = (⌊(657+196883-196560)^(1/3)⌉^3-657)^(1/3)
    343 = ⌊(657+196883-196560)^(1/3)⌉^3-657 = 7^3
    703 = ⌊(657+196883-196560)^(1/3)⌉^3-297
    883 = ⌊(657+196883-196560)^(1/3)⌉^3-117
    spacing
    σ(145) = 180
    σ(323) = 360
    σ(290) = 540
    /5pell function? all numeracy
    2320 = = (2*290+1)^2-(2*(290-1)+1)^2
    2362 = (
    2*(5*59)+1+
    2*((5*59)+(5*59)^2)+
    2*((5*59)+(5*59)^2)+1
    )-(
    2*(5*59-1)+1+
    2*((5*59-1)+(5*59-1)^2)+
    2*((5*59-1)+(5*59-1)^2)+1) = 4724 / 2
    “CR˚0$$ that BRI(p.36) when nigh fined IT” DD
    297 88209 595 177012 177013 0 354025 31333248144 31333602169 593 2 1188 1188 0 2376 419169168 419171544 354620 2378

  163. Paul Vaughan says:

    primorial
    2# = 2
    3# = 3*2 = 6
    5# = 5*3*2 = 30
    7# = 7*5*3*2 = 210

    super-primorial
    2## = 2
    3## = 3#*2# = 12
    5## = 5#*3#*2# = 360 = σ(196883-196560)
    7## = 7#*5#*3#*2# = 75600 = 378*(378-178)

  164. Paul Vaughan says:

    X-pawn save IT
    178 = Σs(44)
    alley mi+c queue
    7## = 378*(378-Σs(44))
    add diss high pell
    32 = Σs(15)
    compute331in co˚too chasm
    32 = Σs(3*5)
    1314 = 1+2+3+12+17+28+32+72+108+117+297+657-Σs(3*5)
    twin prime
    s(33) = 15 = s(16)
    navy blue watt?
    s(15) = 9
    s(9) = 4
    s(4) = 3
    s(3) = 1
    s(1) = 0
    or WellCOunT˚˚7: 1 4 1 2 3 1 4

  165. Paul Vaughan says:

    58 = ⌊196883^(1/3)⌋
    59 = ⌈196883^(1/3)⌉
    117 = ⌊196883^(1/3)⌋ + ⌈196883^(1/3)⌉ = x mod 180 where x = 657, 297, 117
    220 = average(117,323)
    220 = average(floor((47*59*71)^(1/3),1)+ceiling((47*59*71)^(1/3),1),196883-196560)

    R(1,1/2,58) = 0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π
    R(2,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π
    R(3,1/2,58) = 139560.000034332 = ⌊(e^√58π)^(1/3)⌉^3 – e^√58π
    R(4,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π
    104 = d(2,1/2,58) = R(2,1/2,58) – R(1,1/2,58)
    139560 = d(3,1/2,58) = R(3,1/2,58) – R(1,1/2,58)
    104 = d(4,1/2,58) = R(4,1/2,58) – R(1,1/2,58)

    R(1,1/3,59) = 0.0133833873842377 = ⌊(e^π*59^(1/3))^(1/1)⌉^1 – e^π*59^(1/3)
    R(2,1/3,59) = 323.013383387384 = ⌊(e^π*59^(1/3))^(1/2)⌉^2 – e^π*59^(1/3)
    323 = d(2,1/3,59) = R(2,1/3,59) – R(1,1/3,59)

    R(1,1/2,28) = 0.0118744149804115 = ⌊(e^√28π)^(1/1)⌉^1 – e^√28π
    R(2,1/2,28) = 553.01187441498 = ⌊(e^√28π)^(1/2)⌉^2 – e^√28π
    R(3,1/2,28) = 744.01187441498 = ⌊(e^√28π)^(1/3)⌉^3 – e^√28π
    R(4,1/2,28) = 196585.011874415 = ⌊(e^√28π)^(1/4)⌉^4 – e^√28π
    196585 = d(4,1/2,28) = R(4,1/2,28) – R(1,1/2,28)
    744 = d(3,1/2,28) = R(3,1/2,28) – R(1,1/2,28)
    553 = d(2,1/2,28) = R(2,1/2,28) – R(1,1/2,28)

    7 = ⌊323^(1/3)⌉
    14*158 = 7*316 = 4*553 = 2212 anomalistic JEV

    657 = 553+104

    57000 = 196560-139560
    = 3*19*round((657+196883-196560)^(1/3),0)^3
    = 57*⌊980^(1/3)⌉^3

  166. Paul Vaughan says:

    miscellaneous notes

    7## = 378*(378-178)
    7## = 378*(378-φ(290))
    7## = 378*(378-s(225))

    “The only Pell numbers that are squares, cubes, or any higher power of an integer are 0, 1, and 169 = 13^2.”

    worthwhile: explore trivial systematic relations with Pell – e.g. difference, integrate, sum & difference for even & odd indices, look at ratio convergences, differences with increasing index gaps, silver mean (binet gold analog), etc.
    https://oeis.org/A002203
    https://oeis.org/A001333

  167. Paul Vaughan says:

    test
    7 49 15 112 113 0 225 12544 12769 13 2 28 28 0 56 5488 5544 240 58 8 840 294 78 12 97 98 12319 29 28 196 198 195 99 0 30

  168. Paul Vaughan says:

    1692 = 2112420
    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)

  169. Paul Vaughan says:

    abstract appearance of anomalistic hindsight: 58 never lost in the no. stream

    60.8544553085225 = slip(29.4701958106261,11.8627021700857)
    936.955612197393 = slip(60.8544553085225,19.8549641949401)
    937 = average(
    -((2*(234-1)+1)+(2*((234-1)+(234-1)^2))+(2*((234-1)+(234-1)^2)+1))
    ,((2*(234-0)+1)+(2*((234-0)+(234-0)^2))+(2*((234-0)+(234-0)^2)+1)) )

    234 = 2 * 117
    234.238903049342 = slip(65.0814383478416,19.8549641949401)
    117.119451524671 = slip(50.9307341302372,19.8549641949401) ~= 117

    review foundations

    19.8549641949401 = beat(29.4701958106261,11.8627021700857)
    9.92748209747005 = 19.8549641949401 / 2
    4.96374104873503 = 19.8549641949401 / 4

    16.9161211952138 = harmean(29.4701958106261,11.8627021700857)
    8.45806059760692 = axial(29.4701958106261,11.8627021700857)
    4.22903029880346 = 8.45806059760692 / 2
    2.11451514940173 = 8.45806059760692 / 4

    supplementary
    17.2271460578308 = beat(936.955612197393,16.9161211952138)
    8.61357302891541 = 17.2271460578308 / 2
    4.30678651445771 = 17.2271460578308 / 4
    ||||||||||||||||||||||||||||||||||||||||||||||||||||
    33.2322557333676 = harmean(936.955612197393,16.9161211952138)
    16.6161278666838 = axial(936.955612197393,16.9161211952138)
    8.3080639333419 = 16.6161278666838 / 2
    4.15403196667095 = 16.6161278666838 / 4

    130.162876695683 = slip(19.8549641949401,17.2271460578308)
    65.0814383478416 = slip(19.8549641949401,8.61357302891541)
    50.9307341302372 = slip(19.8549641949401,4.30678651445771)
    ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
    101.861468260475 = slip(19.8549641949401,16.6161278666838)
    50.9307341302375 = slip(19.8549641949401,8.3080639333419)
    90.1207922119932 = slip(19.8549641949401,4.15403196667095)
    ————————————————————————————
    alternate derivation
    6.57189983390616 = axial(14.735097905313,11.8627021700857)
    13.1437996678123 = harmean(14.735097905313,11.8627021700857)

    10.7471650296459 = beat(16.9161211952138,6.57189983390616)
    5.37358251482297 = 10.7471650296459 / 2
    2.68679125741149 = 10.7471650296459 / 4
    ||||||||||||||||||||||||||||||||||||||||||||||||||||
    9.46619163320184 = harmean(16.9161211952138,6.57189983390616)
    4.73309581660092 = axial(16.9161211952138,6.57189983390616)
    2.36654790830046 = 4.73309581660092 / 2
    1.18327395415023 = 4.73309581660092 / 4

    130.162876695683 = slip(19.8549641949401,10.7471650296459)
    65.0814383478414 = slip(19.8549641949401,5.37358251482297)
    50.9307341302375 = slip(19.8549641949401,2.68679125741149)
    ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
    101.861468260475 = slip(19.8549641949401,4.73309581660092)
    50.9307341302375 = slip(19.8549641949401,2.36654790830046)
    90.1207922119933 = slip(19.8549641949401,1.18327395415023)
    ————————————————————————————
    further review
    2362.08778401397 = slip(101.861468260475,8.45806059760692)
    2362.08778401782 = slip(50.9307341302372,8.45806059760692)
    4724.17556802794 = slip(101.861468260475,16.9161211952138)
    4724.17556803565 = slip(50.9307341302372,16.9161211952138)

  170. Paul Vaughan says:

    llucklly enough
    1 = (1+3) / 4
    2 = (1+7) / 4
    3 = (1+11) / 4
    5 = (1+19) / 4
    11 = (1+43) / 4
    sum = 22 = 1*10 + 12 ; 11 = 1*5+3#
    sum = 58 = 7*10 12 ; 29 = 7*5-3# ; 12 = 3##
    17 = (1+67) / 4
    41 = (1+163) / 4
    70+22-10-58 = 24 ; so average
    70+44-10-58 = 104-58 = average(22,70) = Φ(47) ; hhear in the˚Center
    478 = 2 * 239 = 58+420
    478 = sum 744 levels + sum 104 levels
    478 = 28+163+67+43+19 + 10+13+18+22+37+58
    420 = 28+163+67+43+19 + 10+13+18+22+37 = 316+104
    no. how to spell functionllnumeracy
    490 = 12+70+408 = 58 + 432 = 70 + 420
    with token weather part III or 2D bait
    58 3364 117 6844 6845 0 13689 46840336 46854025 115 2 232 232 0 464 3121792 3122256 13806 466 8 400374 20184 690 12 6727 6728 46826647 233 232 13456 13458 13455 6729 0 234
    -((2*(7-1)+1)+(2*((7-1)+(7-1)^2))+(2*((7-1)+(7-1)^2)+1))
    +((2*(7-0)+1)+(2*((7-0)+(7-0)^2))+(2*((7-0)+(7-0)^2)+1)) = 58
    10000 = 100^2 ; (23) 1 4 1 2 3 1 4 (29) ; (58) 1 4 1 2 3 1 4 (70)
    100 = 10+13+18+22+37 = sum 1st 23 primes = 2212-2112
    158 = 10+13+18+22+37+58
    432 186624 865 374112 374113 0 748225 1.3996E+11 1.39961E+11 863 2 1728 1728 0 3456 1289945088 1289948544 749090 3458 8 161803440 1119744 5178 12 373247 373248 1.39959E+11 1729
    -((2*(432-1)+1)^2)
    +((2*(432-0)+1)^2) = 3456 =
    -((2*((432-1)+(432-1)^2))+(2*((432-1)+(432-1)^2)+1))
    +((2*((432-0)+(432-0)^2))+(2*((432-0)+(432-0)^2)+1))
    add JSUN type won near ice awe sol ease try angles
    -((2*(432-1)+1)+(2*((432-1)+(432-1)^2))+(2*((432-1)+(432-1)^2)+1))
    +((2*(432-0)+1)+(2*((432-0)+(432-0)^2))+(2*((432-0)+(432-0)^2)+1)) =
    3458 = 2112+1346

    1346 = 1+2+3+12+17+28+32+72+108+117+297+657

    12 = 432-420 = 490-478 = (19+43+67)-117 ; 24 = 1+2+3+7+11 = 71-47
    without missinllink, well above, said taco to X Pence Sov. snow “don˚T no.”
    (19+43+67)-71 = 117-59 = 58 ; 104 = 163-59 ; 117 = 163-Φ(47)
    σ(Φ(47))^5 – Φ(47)^5 – 47^5 = 19^5 + 43^5 + 67^5 ; Σφ(323) = 47

  171. Paul Vaughan says:

    (23) 1 4 1 2 3 1 4 (29)
    (58) 4 1 3 2 1 4 1 (70) typo˚correction
    2022.00011048077 =
    slip(slip(slip(19.8549641949401,1.18327395415023),8.45806059760692),slip(19.8549641949401,2.36654790830046))

    Φ(59) = 58
    Φ(58) = 28
    Φ(28) = 12
    Φ(12) = 4
    Φ(4) = 2
    Φ(2) = 1 ; 105 = ΣΦ(59)

    -104+Φ(163) = 163-ΣΦ(59) = Φ(59) = 58 = -ΣΦ(59)+ΣΦ(378) ; ΣΦ(378) = 163 = 104+59
    √(1^2+2^2+3^2+…+22^2+23^2+24^2) = 70 = 104-34 ; 34 = average(10,58) = s(6)+s(28)

  172. Paul Vaughan says:

    index prime
    1 2
    2 3
    3 5 —
    4 7 ——

    5 11
    7 17 ——

    17 59

    117 = Φ(59)+59
    Party in the ˚Center
    24 = 19+43+67-ΣΦ(59) = (70-10)-(58-22) ; 1184 = s(1210) ; 894 = Σδ(902)
    √(1^2+2^2+3^2+…+22^2+23^2+24^2) = 70 = 117-47 ; Φ(47) = 117-71
    117 = Φ(59) + 59 ; ΣΦ(59) = Φ(59)+47 ; 71+Φ(59) = 19+43+67
    2112 = 2432-28-163-67-43-19 = 744+1368 = 1210 + 902 = 24^2 + 1536
    nos. by (s(n)) amicably enough
    902 = Σδ(894) = Σδ(Σδ(993)) = Σδ(Σδ(Σδ(378))) ; 1210 = s(1184)
    41 = 64-23 = 70-29 ; 47 = 70-23 = 117-70
    3# – 29-23 = 64-58 = 70-64 = 47-41 = average(19+43+67,-ΣΦ(59))
    (23) 1 4 1 2 3 1 4 (29)
    (58) 4 1 3 2 1 4 1 (64) another typo˚correction
    (64) 4 1 3 2 1 4 1 (70)
    28 = 7+2+5+3+2+7+2 = 11+17 = s(28)
    sum of primes indexed by sequence of 7 he[luckily]ignore
    prime minus index
    3 = 7-4
    2 = 5-3 ; 2^2 = 5^2 – 3^2 — lowest twin prime PPT hyp010use piece 5=2(v)+3(^)
    1 = 3-2

    3^1
    3^2 + 4^2 = 5^2
    3^3 + 4^3 + 5^3 = 6^3
    3^4 + 4^4 + 5^4 + 6^4 = 2258 = 2 * 1129

  173. Paul Vaughan says:

    Net search doesn’t find (mystery indeed) 2 key relationships established above (1 of which is generalizable from the special case noted above to all cases).

  174. Paul Vaughan says:

    340 results found: https://oeis.org/search?q=1+2+3+1+4
    127 results found: https://oeis.org/search?q=4+1+3+2+1
    lucky seek winshe ignore˚C0[MP]PR($(n))Sh!Un…with no. scandal

    1
    1 = 1 * 1
    2 = 1 * 2
    6 = 2 * 3
    6 = 6 * 1
    24 = 6 * 4

    1
    2 = 1 + 1
    3 = 1 + 2
    5 = 2 + 3
    11 = 6 + 5
    17 = 6 + 11
    41 = 24 + 17

  175. Paul Vaughan says:

    00 note
    even write

    if
    0 =
    A000534(n-1)-A025428(n-1) =
    A000534(n-0)-A025428(n-0)
    then
    n = A296579(i)
    and
    difference = 1

    240-1 = 239 = average(158,320)
    368-1 = 367 = 178+189 ; 189 * 2 = 378
    448-1 = 447 = ΣΦ(323)

    Σδ(378) = 752 = 1504 / 2
    Σδ(752) = 894
    Σδ(894) = 902 ; Σδ(902) = 894 = 2 * 447

    1
    376 = 1*376

    1
    2 = 1+1
    378 = 376+2

  176. Paul Vaughan says:

    in other words
    “Number of partitions of n into 3 squares (allowing part zero).”
    https://oeis.org/A000164
    note instances of 2 consecutive 0s: https://oeis.org/A000164/b000164.txt
    735 = 367+368 = 368^2-367^2 = ΣΦ(1800) ; 1800 = σ(894)
    key insight arising from (luckily he ignore) “the butterfly expansion” of A025428:
    189 = 367-178 ; then countless insights stream – too many to note all at once – select an example
    188 = 368-180 ; 180 = σ(145) = 297-117
    adjacent type 2 PPTs
    1500 = 4*188^2-4*187^2
    1500 = (4*188)^2-(4*187)^2 = 752^2-748^2
    whereas
    184 = average(180,188)
    1470 = average(
    -( (4*183) + ((4*183^2)-1) + ((4*183^2)+1) )
    ,( (4*184) + ((4*184^2)-1) + ((4*184^2)+1) ) )
    with type 1 PPT
    735 = 2*367+1 = side
    otherside & hypotenuse change from previous in series
    1468 = (2*(367+367^2)) – (2*(366+366^2)) = (2*(367+367^2)+1) – (2*(366+366^2)+1)
    property of next triangle in series (case m = 368) compared with previous (case m = 367)
    735 = 368^2-367^2
    1472 = (2*(368+368^2)) – (2*(367+367^2)) = (2*(368+368^2)+1) – (2*(367+367^2)+1)
    mean side change from previous in type 1 PPT series for pair m = 367 & m = 368
    1470 = average(1468,1472)

  177. Paul Vaughan says:

    no. typo mystery:
    6000 = (4*188)^2-(4*187)^2 = 752^2-748^2

    2nd lowest amicable pair: 1184 = s(1210) & 1210 = s(1184)

    butterfly expansion A025428
    1st (& only in 1st 10k) occurence of 4th power preceding square is for 2368 = 2*1184
    81 = 3^4 precedes 3^2 = 9

    The exceptional case resolves (natural & whole number fractal) focus.

    type 1 PPT
    295 87025 591 174640 174641 0 349281 30499129600 30499478881 589 2 1180 1180 0 2360 410758000 410760360 349872 2362 8 51606120 522150 3534 12 174049 174050 30498780319 1181 295 1180 348100 348102 348099 174051 0 1182

    296 87616 593 175824 175825 0 351649 30914078976 30914430625 591 2 1184 1184 0 2368 414949376 414951744 352242 2370 8 52131816 525696 3546 12 175231 175232 30913727327 1185 296 1184 350464 350466 350463 175233 0 1186

    297 88209 595 177012 177013 0 354025 31333248144 31333602169 593 2 1188 1188 0 2376 419169168 419171544 354620 2378 8 52661070 529254 3558 12 176417 176418 31332894119 1189 297 1188 352836 352838 352835 176419 0 1190

  178. Paul Vaughan says:

    Average Party
    in the Center

    63 = 22+18+13+10
    79 = average(63,95) = 158 / 2
    95 = 37+58
    158 = 95+63 = 2 * 79

    1346 = 1+2+3+12+17+28+32+72+108+117+297+657 = 1314 + 32 ; notice where 47 isn’t

    top*left-1
    _ 8 16 32 64 128
    1 7 15 31 63 127
    2 15 31 63 127 255
    3 23 47 95 191 383
    4 31 63 127 255 511
    5 39
    79 159 319 639
    6 47 95 191 383 767
    7 55 111 223 447 895
    8 63 127 255 511 1023
    9 71 143 287 575 1151
    10 79 159 319 639 1279
    11 87 175 351 703 1407
    12 95 191 383 767 1535
    13 103 207 415 831 1663
    14 111 223 447 895 1791
    15 119 239 479 959 1919
    16 127 255 511 1023 2047

    left mod top
    _ 8 16 32 64 128 256
    127 7 15 31 63 127 127
    111 7 15 15 47 111 111
    79 7 15 15 15 79 79
    47 7 15 15 47 47 47
    15 7 15 15 15 15 15

    00 note even write makes cumulative difference

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

    111
    112 1 1
    239 127 128
    240 1 129 1
    367 127 256 128
    368 1 257 129 1
    447 79 336 208 80
    448 1 337 209 81 1
    495 47 384 256 128 48
    496 1 385 257 129 49
    623 127 512 384 256 176
    624 1 513 385 257 177
    751 127 640 512 384 304
    752 1 641 513 385 305
    879 127 768 640 512 432
    880 1 769 641 513 433
    959 79 848 720 592 512
    960 1 849 721 593 513
    1007 47 896 768 640 560
    1008 1 897 769 641 561
    1135 127 1024 896 768 688
    1136 1 1025 897 769 689
    1263 127 1152 1024 896 816
    1264 1 1153 1025 897 817
    1391 127 1280 1152 1024 944
    1392 1 1281 1153 1025 945
    1471 79 1360 1232 1104 1024
    1472 1 1361 1233 1105 1025
    1519 47 1408 1280 1152 1072
    1520 1 1409 1281 1153 1073
    1647 127 1536 1408 1280 1200
    1648 1 1537 1409 1281 1201
    1775 127 1664 1536 1408 1328
    1776 1 1665 1537 1409 1329
    1791 15 1680 1552 1424 1344
    1792 1 1681 1553 1425 1345
    1903 111 1792 1664 1536 1456
    1904 1 1793 1665 1537 1457
    1983 79 1872 1744 1616 1536
    1984 1 1873 1745 1617 1537

    incorrect link above – correction:
    σ(Φ(47))^5 – Φ(47)^5 – 47^5 = 19^5 + 43^5 + 67^5

    936.955612197409 = 1/(-2/11.8627021700857+5/29.4701958106261)
    4270.51884168654 = 2/(1/11.8627021700857-3/29.4701958106261+1/84.0331316671926+1/164.793624044745)
    anomalistic periods (Standish 1992)
    1536.74746987137 = harmean(4270.51884168654,936.955612197409)

  179. Paul Vaughan says:

    Be aware there are some (unnoted) typos in recent comments (a cost of necessary time management). Check carefully.

  180. Paul Vaughan says:

    at the center 4 dull typos (includes h/tml tags above)

    say something boring

    70-inclusive 1st 2 forward butterfly expansion A025428

    58	4	1	2	6	10	22	46	70	166	262	358	646	1510	3238	6694	20518	48166
    59	1	1	2	3	6	12	18	42	66	90	162	378	810	1674	5130	12042	18954
    60	3	1	2	5	11	17	41	65	89	161	377	809	1673	5129	12041	18953	39689
    61	2	1	2	4	6	14	22	30	54	126	270	558	1710	4014	6318	13230	27054
    62	1	1	2	3	7	11	15	27	63	135	279	855	2007	3159	6615	13527	34263
    63	4	1	2	6	10	14	26	62	134	278	854	2006	3158	6614	13526	34262	117206
    64	1	1	2	3	4	7	16	34	70	214	502	790	1654	3382	8566	29302	70774
    

    Heegner’s lucky
    2 reverse butterfly expansion A025428

    58	4	1	2	6	14	14	14	14	14	14	14	14	14	14	14	14	14	14	14
    59	1	1	2	3	7	15	15	15	15	15	15	15	15	15	15	15	15	15	15
    60	3	1	2	5	8	20	44	44	44	44	44	44	44	44	44	44	44	44	44
    61	2	1	2	4	10	16	40	88	88	88	88	88	88	88	88	88	88	88	88
    62	1	1	2	3	5	11	17	41	89	89	89	89	89	89	89	89	89	89	89
    63	4	1	2	6	10	18	42	66	162	354	354	354	354	354	354	354	354	354	354
    64	1	1	2	3	7	11	19	43	67	163	355	355	355	355	355	355	355	355	355
    
  181. Paul Vaughan says:

    opaque (at the center for average insight)

    forward butterfly expansion
    beginning on every line of A138967
    (luckily too) includes Heegner nos.
    ( same for reverse construction (n-2)(n-1) instead of (n-1)(n-2) (a type of symmetry) )

    also given above: solution to OEIS A100570

  182. Paul Vaughan says:

    Standish (1992) sidereal (to be neither confused nor conflated with anomalistic)
    1806.17637816713 = slip(65.4656719965873,6.57038184300286)

    analogously: compare Seidelmann sidereal (~1800) with tropical (~900) * 2

  183. Paul Vaughan says:

    NASA “factsheet”
    1798.25390532374 = slip(65.4646083498784,6.57038000192319) — sidereal
    899.995342082165 = slip(66.1360443524991,6.56535897749883) — tropical
    Seidelmann (1992)
    1800.67427686801 = slip(65.4650142672071,6.57038862222412) — synodic
    899.727736185918 = slip(66.1361966326052,6.56535973915137) — tropical

  184. Paul Vaughan says:

    ^(1/2) = √

    R(1,1/2,9) = 0.352192083306363 = ⌊(e^√9π)^(1/1)⌉^1 – e^√9π
    R(2,1/2,9) = -70.6478079166936 = ⌊(e^√9π)^(1/2)⌉^2 – e^√9π
    R(3,1/2,9) = -224.647807916694 = ⌊(e^√9π)^(1/3)⌉^3 – e^√9π
    R(4,1/2,9) = 2249.35219208331 = ⌊(e^√9π)^(1/4)⌉^4 – e^√9π
    71 = -d(2,1/2,9) = R(1,1/2,9) – R(2,1/2,9)
    225 = -d(3,1/2,9) = R(1,1/2,9) – R(3,1/2,9) = 15^2
    2249 = d(4,1/2,9) = R(4,1/2,9) – R(1,1/2,9)
    2320 = 2249 + 71 = ⌊(e^3π)^(1/4)⌉^4 – ⌊(e^3π)^(1/2)⌉^2
    = round(exp(3*pi())^(1/4),0)^4 – round(exp(3*pi())^(1/2),0)^2

  185. Paul Vaughan says:

    15^2 = 225

    5090 = ( 71 + 225 + 2249 ) * 2 = 2 * ( 2320 + 225 ) = ΣΣδ(220) * 2
    2545 = 71 + 225 + 2249 = 2320 + 15^2 = 2320 + 225 = ΣΣδ(220)
    2545 = R(1,1/2,9) – R(2,1/2,9) – R(3,1/2,9) + R(4,1/2,9)
    2545 = round(exp(3*pi())^(1/1),0)^1 – round(exp(3*pi())^(1/2),0)^2 – round(exp(3*pi())^(1/3),0)^3 + round(exp(3*pi())^(1/4),0)^4
    5090 = ( R(1,1/2,9) – R(2,1/2,9) – R(3,1/2,9) + R(4,1/2,9) ) * 2
    5090 = ( round(exp(3*pi())^(1/1),0)^1 – round(exp(3*pi())^(1/2),0)^2 – round(exp(3*pi())^(1/3),0)^3 + round(exp(3*pi())^(1/4),0)^4 ) * 2

    1800 ~= 2545 / √2 ~= 1 / ( 1 / 1470 + 1 / 2320 )
    980 = σ(178+378) = 657+323

  186. Paul Vaughan says:

    104-levels 10 & 18: powers 2 & 4

    R(1,1/2,10) = 0.212132286567794 = ⌊(e^√10π)^(1/1)⌉^1 – e^√10π
    R(2,1/2,10) = 104.212132286568 = ⌊(e^√10π)^(1/2)⌉^2 – e^√10π
    R(3,1/2,10) = -948.787867713432 = ⌊(e^√10π)^(1/3)⌉^3 – e^√10π
    R(4,1/2,10) = 104.212132286568 = ⌊(e^√10π)^(1/4)⌉^4 – e^√10π
    104 = d(2,1/2,10) = R(2,1/2,10) – R(1,1/2,10)
    949 = -d(3,1/2,10) = R(1,1/2,10) – R(3,1/2,10)
    104 = d(4,1/2,10) = R(4,1/2,10) – R(1,1/2,10)
    845 = 949 – 104
    = 2*round(exp(sqrt(10)*pi())^(1/1),0)^1 – round(exp(sqrt(10)*pi())^(1/3),0)^3 – round(exp(sqrt(10)*pi())^(1/2),0)^2
    = 2*round(exp(sqrt(10)*pi())^(1/1),0)^1 – round(exp(sqrt(10)*pi())^(1/3),0)^3 – round(exp(sqrt(10)*pi())^(1/4),0)^4

    R(1,1/2,18) = 0.00711438176222146 = ⌊(e^√18π)^(1/1)⌉^1 – e^√18π
    R(2,1/2,18) = 104.007114381762 = ⌊(e^√18π)^(1/2)⌉^2 – e^√18π
    R(3,1/2,18) = -426.992885618238 = ⌊(e^√18π)^(1/3)⌉^3 – e^√18π
    R(4,1/2,18) = 104.007114381762 = ⌊(e^√18π)^(1/4)⌉^4 – e^√18π
    104 = d(2,1/2,18) = R(2,1/2,18) – R(1,1/2,18)
    427 = -d(3,1/2,18) = R(1,1/2,18) – R(3,1/2,18)
    104 = d(4,1/2,18) = R(4,1/2,18) – R(1,1/2,18)
    323 = 427 – 104
    = 2*R(1,1/2,18) – R(3,1/2,18) – R(2,1/2,18)
    = 2*round(exp(sqrt(18)*pi())^(1/1),0)^1 – round(exp(sqrt(18)*pi())^(1/3),0)^3 – round(exp(sqrt(18)*pi())^(1/2),0)^2
    = 2*R(1,1/2,18) – R(3,1/2,18) – R(4,1/2,18)
    = 2*round(exp(sqrt(18)*pi())^(1/1),0)^1 – round(exp(sqrt(18)*pi())^(1/3),0)^3 – round(exp(sqrt(18)*pi())^(1/4),0)^4

    R(1,1/3,59) = 0.0133833873842377 = ⌊(e^π*59^(1/3))^(1/1)⌉^1 – e^π*59^(1/3)
    R(2,1/3,59) = 323.013383387384 = ⌊(e^π*59^(1/3))^(1/2)⌉^2 – e^π*59^(1/3)
    323 = d(2,1/3,59) = R(2,1/3,59) – R(1,1/3,59)
    = round(exp(59^(1/3)*pi())^(1/2),0)^2 – round(exp(59^(1/3)*pi())^(1/1),0)^1
    323 = 196883 – 196560

    5## = σ(323)
    (5#)^2 = 900

  187. Paul Vaughan says:

    R(1,1/2,5) = -0.186262891398655 = ⌊(e^√5π)^(1/1)⌉^1 – e^√5π
    R(2,1/2,5) = 31.8137371086013 = ⌊(e^√5π)^(1/2)⌉^2 – e^√5π
    R(3,1/2,5) = -124.186262891399 = ⌊(e^√5π)^(1/3)⌉^3 – e^√5π
    R(4,1/2,5) = 171.813737108601 = ⌊(e^√5π)^(1/4)⌉^4 – e^√5π
    32 = d(2,1/2,5) = R(2,1/2,5) – R(1,1/2,5) = s(58) = 1346 – 1314
    124 = -d(3,1/2,5) = R(1,1/2,5) – R(3,1/2,5)
    172 = d(4,1/2,5) = R(4,1/2,5) – R(1,1/2,5)
    140 = 172 – 32 = ⌊(e^√5π)^(1/4)⌉^4 – ⌊(e^√5π)^(1/2)⌉^2
    = round(exp(sqrt(5)*pi())^(1/4),0)^4 – round(exp(sqrt(5)*pi())^(1/2),0)^2
    140: 1 2 4 5 7 10 14 20 28 35 70 140
    5 = harmean(1,2,4,5,7,10,14,20,28,35,70,140)
    5 = harmean(1,2,4,8,16,31,62,124,248,496)

    496: 1 2 4 8 16 31 62 124 248 496
    496 = 4 * 124 = -4 * d(3,1/2,5)
    124 = ⌊(e^√5π)^(1/1)⌉^1 – ⌊(e^√5π)^(1/3)⌉^3
    = round(exp(sqrt(5)*pi())^(1/1),0)^1 – round(exp(sqrt(5)*pi())^(1/3),0)^3
    496 = 4 * ( ⌊(e^√5π)^(1/1)⌉^1 – ⌊(e^√5π)^(1/3)⌉^3 )

  188. Paul Vaughan says:

    117 = -63+180*2^0 ; 0+1 = 1
    297 = -63+180*2^1 ; 1+1 = 2
    657 = -63+180*2^2 ; 2+1 = 3

    194 = σ(p(19)-p(17)) where p(x) denotes partition function of x

  189. Paul Vaughan says:

    17

    columns
    1: x
    2: d(2,1/2,x)=R(2,1/2,x)-R(1,1/2,x)=⌊(e^√xπ)^(1/2)⌉^2-e^√xπ-⌊(e^√xπ)^(1/1)⌉^1
    3: R(2,1/2,x)=⌊(e^√xπ)^(1/2)⌉^2-e^√xπ
    4: R(1,1/2,x)=⌊(e^√xπ)^(1/1)⌉^1-e^√xπ

    72 8744 8744.31331302 0.31331302
    88 8744 8744.07673348 0.07673348
    100 8744 8743.9709886 -0.0290114 ————- 100/4 = 25 = 5^2 note: 152 not 104
    148 8744 8743.999 -0.001
    232 8744 8744 0

    640=72+88+148+232 + 100
    540=72+88+148+232

  190. Paul Vaughan says:

    typo
    no “-e^√xπ” in equation for column 2:
    d(2,1/2,x)=R(2,1/2,x)-R(1,1/2,x)=⌊(e^√xπ)^(1/2)⌉^2-e^√xπ-⌊(e^√xπ)^(1/1)⌉^1
    correction
    d(2,1/2,x)=R(2,1/2,x)-R(1,1/2,x)=⌊(e^√xπ)^(1/2)⌉^2-⌊(e^√xπ)^(1/1)⌉^1
    _
    640=744-104

  191. Paul Vaughan says:

    18 = 180 / 10 = 29-11 = average(-29,65)
    18 = 1+2+3+12
    35 = 1+2+3+12+17 = 13 + 22
    63 = 1+2+3+12+17+28 = 10 + 13 + 18 + 22
    95 = 1+2+3+12+17+28+32 = 37 + 58

    158 = 10+13+18+22+37+58
    553 = 657-104
    1106 = 158*7 = 553*2
    2212 = 158*14
    4424 = 158*28 ; 28 = s(28)

    29 = (3^4+4^3)/5
    145 = 3^4+4^3
    σ(145) = 1+5+29+145 = 180
    σ(290) = 1+2+5+10+29+58+145+290 = 540
    540 – 180 = 360 = σ(323) ; 323 = 17*19 = -196560+196883 ; 196883 = 47*59*71

    29 = (4^3+3^4)/5
    145 = 4^3+3^4
    145 = 8^2+9^2
    245 = 8^2+9^2+10^2
    490 = p(19)
    297 = p(17)
    194 = σ(p(19)-p(17)) ; 193 = 490-297

    36 = 19+17 = (1+2+3)^2 = average(3,5)*gcd(117,297,657)
    18 = (19+17)/(19-17) = average(19,17)
    9 = gcd(117,297,657) = average(19,17)/(19-17)
    4 = average(3,5) = 2^2
    2 = 19-17 = 5-3

  192. Paul Vaughan says:

    290=5 * 58
    15 = 5 * 3 ; 55 = 58 – 3 = 5^2 + 4^2 + 3^2 + 2^2 + 1^2

    744 = (290-104)*4 = 27^2 + 15
    178 = (104-15)*2

    R(1,1/2,58) = 0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π
    R(2,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π
    R(3,1/2,58) = 139560.000034332 = ⌊(e^√58π)^(1/3)⌉^3 – e^√58π
    R(4,1/2,58) = 104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π
    104 = d(2,1/2,58) = R(2,1/2,58) – R(1,1/2,58)
    139560 = d(3,1/2,58) = R(3,1/2,58) – R(1,1/2,58) —————————————
    104 = d(4,1/2,58) = R(4,1/2,58) – R(1,1/2,58)

    1	15	15	225	_	15				
    2	15	14	196	_	14	1			
    3	15	26	676	_	13	2			
    4	15	36	1296	_	12	3			
    5	15	44	1936	_	11	4			
    6	15	50	2500	_	10	5			
    7	15	54	2916	_	9	6			
    8	15	56	3136	_	8	7			
    9	15	24	576	_	12	2	1		
    10	15	33	1089	_	11	3	1		
    11	15	40	1600	_	10	4	1		
    12	15	45	2025	_	9	5	1		
    13	15	48	2304	_	8	6	1		
    14	15	60	3600	_	10	3	2		
    15	15	72	5184	_	9	4	2		
    16	15	80	6400	_	8	5	2		
    17	15	84	7056	_	7	6	2		
    18	15	96	9216	_	8	4	3		
    19	15	105	11025	_	7	5	3		
    20	15	120	14400	_	6	5	4		
    21	15	54	2916	_	9	3	2	1	
    22	15	64	4096	_	8	4	2	1	
    23	15	70	4900	_	7	5	2	1	
    24	15	84	7056	_	7	4	3	1	
    25	15	90	8100	_	6	5	3	1	
    26	15	144	20736	_	6	4	3	2	
    27	15	120	14400	_	5	4	3	2	1
    n	sum	product	product^2	_	distinct parts				
    _	_	_	139560	= sum(product^2)					
    
  193. Paul Vaughan says:

    1 = 22-18-13+10
    2 = 10/(18-13)
    3 = 13-10
    4 = 22-18
    5 = 18-13
    6 = 18/(13-10)
    12 = (22-18)*(13-10)
    17 = 22+13-18
    28 = 18+10
    32 = 58+37-22-18-13-10
    72 = 58+37-13-10
    108 = 58+37+13
    145 = 58+37+22+18+10
    29 = ((13-10)^(22-18)+(22-18)^(13-10))/(18-13) = (3^4+4^3)/5
    145 = (13-10)^(22-18)+(22-18)^(13-10) = 3^4+4^3
    180 = σ(145) = 1+5+29+145
    =(22-18-13+10)
    +(18-13)
    +((13-10)^(22-18)+(22-18)^(13-10))/(18-13)
    +(13-10)^(22-18)+(22-18)^(13-10)
    63 = 22+18+13+10
    117 = -63+180*2^0
    297 = -63+180*2^1
    657 = -63+180*2^2

  194. Paul Vaughan says:

    _ 1 2 3 7 11 19
    1
    2 3
    3 4 5
    7 8 9 10
    11 12 13 14 18
    19 20 21 22 26 30

  195. Paul Vaughan says:

    supplementary
    29 = √(21^2+20^2)
    12 = 3 * 4 = ((13-10)*(18-13)/5) * ((18-13)*(22-18)/5) = (22-18)*(37-22)/5
    8 & 9 are the only perfect powers 1 apart
    145 = 64 + 81 = 8^2 + 9^2
    245 = 64 + 81 +100 = 8^2 + 9^2 + 10^2

    |⌊(e^√(above)π)^(1/2)⌉^2 – ⌊(e^√(above)π)^(1/1)⌉^1|
    _ 1 2 3 7 11 19
    1
    2 6
    3 6 32
    7 3 71 104
    11 109 104 90 104
    19 419 391 104 2862 3058

    |104-above|
    98
    98 72
    101 33 0
    5 0 14 0
    315 287 0 2758 2954
    315 = (58-37)*(37-22) = 5*(10+13+18+22)

    14: Heegner index 3,5
    +104

    exp and 1,2,3
    H(3)*H(n) n=4..8
    3*7 = 21 = 58-37 = (10+13+18+22)/3
    3*11 = 33 = 104-71
    3*19 = 57 = 104-47
    3*43 = 129; s(129) = 47
    3*67 = 201; s(201) = 71
    note selection
    163+67 = 608-378
    163*4 = s(608) = 652
    s(652) = 496 = s(496)

  196. Paul Vaughan says:

    simply differencing again and again
    10
    13 3
    18 5 2
    22 4 -1 -3
    37 15 11 12 15
    58 21 6 -5 -17 -32 =
    (((((58-37)-(37-22))-((37-22)-(22-18)))-(((37-22)-(22-18))-((22-18)-(18-13))))-((((37-22)-(22-18))-((22-18)-(18-13)))-(((22-18)-(18-13))-((18-13)-(13-10))))) =
    -1*10
    +5*13
    -10*18
    +10*22
    -5*37
    +1*58
    = 1314 – 1346 = -Σs(15) = -s(58)

    repeat difference

    1	-1	1	-1	1	-1	1	-1	1	-1	1
    0	1	-2	3	-4	5	-6	7	-8	9	-10
    0	0	1	-3	6	-10	15	-21	28	-36	45
    0	0	0	1	-4	10	-20	35	-56	84	-120
    0	0	0	0	1	-5	15	-35	70	-126	210
    0	0	0	0	0	1	-6	21	-56	126	-252
    0	0	0	0	0	0	1	-7	28	-84	210
    0	0	0	0	0	0	0	1	-8	36	-120
    0	0	0	0	0	0	0	0	1	-9	45
    0	0	0	0	0	0	0	0	0	1	-10
    0	0	0	0	0	0	0	0	0	0	1
    

    absolute values for repeat sum (pascal’s triangle)

    3*5 = 15 = -1*22+1*37 =
    +1*10
    -4*13
    +6*18
    -4*22
    +1*37

    104 hindsight: (3,5) PPT

  197. Paul Vaughan says:

    html tag typo correction:
    top*left-1
    _ 8 16 32 64 128
    1 7 15 31 63 127
    2 15 31 63 127 255
    3 23 47 95 191 383
    4 31 63 127 255 511
    5 39 79 159 319 639

    last comparably delightful insight: equation 3
    the 127, 111, 79, 47, 15 pattern is noted in a very different manner on conventional math webpages (as if to deliberately obfuscate – maybe just unhelpfully obsessive devotion to formal convention)
    hindsight: developing number theory independently is feasible
    link noted while cross-referencing (will be of interest to some talkshop readers)
    https://en.wikipedia.org/wiki/600-cell

    deeper awareness of number theory helps
    sort & classify orbital patterns (e.g. anomalistic JEV clearly features 58 & 111)
    scrutinize numerical methods (long list of questions raised (understatement))
    generalize (beyond climate)

    next several comments may appear abstract initially – and later with hindsight maybe not

  198. Paul Vaughan says:

    This graphic from the 600-cell page is reminiscent of past discussions (from back in the nicer days before the politics devolved into weird psy-ops that make no sense) :

  199. Paul Vaughan says:

    repeat difference
    553 = – 1*1 + 8*2 – 28*3 + 56*12 – 70*17 + 56*28 – 28*32 + 8*72 – 1*108

    Prime Sort of Indicator

    240 =
    (( – 1*1 + 5*2 – 10*12 + 10*28 – 5*117 + 1*657 ) +
    ( – 1*3 + 5*17 – 10*32 + 10*72 – 5*108 + 1*297 ))
    /
    (( – 1*1 + 5*2 – 10*12 + 10*28 – 5*117 + 1*657 )
    ( – 1*3 + 5*17 – 10*32 + 10*72 – 5*108 + 1*297 ))

  200. Paul Vaughan says:

    pascal Σs(pell) with prime sort of indicator

    239 = -1*3+5*17-10*32+10*72-5*108+1*297
    240 = Σs(239) = average( 239 , 241 )
    241 = -1*1+5*2-10*12+10*28-5*117+1*657

    Σs(169) = 209 = 836 / 4
    Σs(209) = 241
    Σs(241) = 242
    Σs(242) = 400 = 20^2 = 29^2-21^2 = 447-47
    29 = (3^4+4^3)/5 = average(-3-4-5,20+21+29)

    494 = 894-400 = ΣΦ(323)+Σφ(323)
    408 = 902-494 ; ΣΦ(323) = 447 ; 47 = Σφ(323)
    400 = 894-494 = ΣΦ(323)-Σφ(323)

    20+21+29 = 70 ; Σs(70) = average(178,378) = 278 ; Σs(278) = 628 = 2 * 314
    239 = average(70,408) = average(158,320)

    Σs(239) = 240 ; Σs(240) = 100836 = ⌊657√φ⌉+100k
    163 = 240 + 323 + Σφ(323) – ΣΦ(323) = ΣΦ(323) – 284

  201. Paul Vaughan says:

    15 = 3 * 5
    Σs(15) = 32
    Σs(32) = 64
    Σs(64) = 169
    Σs(169) = 209 —- above this line is broader context
    Σs(209) = 241
    Σs(241) = 242
    Σs(242) = 400 = 21^2 – 41 = s(2401) = σ(7^3) ; σ(σ(400)) = 993

    lowest 744 levels
    repeat difference (pascal)
    19
    28 47 ——————– 28 = s(28) isn’t Heegner but (note well) is 744
    43 71 118 = 2*59

    58th no. with round divisor average = 89 = 178 / 2
    129th no. with round divisor average = 189 = 378 / 2
    129 = 19+43+67
    71 = 129 – 58

    177 = 47+59+71
    σ(177) = 240 ; σ(240) = 744
    s(177) = s(41) = s((163+1)/4) = σ(32) = σ(Σs(15)) = 10+13+18+22 = 63 ; σ(63) = 104
    3,5 twin prime pell PPT spacing

    σ(104) = 7# = 210 ; σ(210) = 576 = 24^2

  202. Paul Vaughan says:

    typo last comment: repeat sum (not difference)
    s(58) = 32 = ΣΣφ(28)
    σ(s(58)) = σ(32) = σ(Σs(15)) = 10+13+18+22
    = s(194)-s(323) = Σs(Σs(37))-37 = 63 ; σ(63) = 104 = 657-553 ; Σδ(553) = 169
    s(323) = 37
    s(194) = 100 = Σs(Σs(37)) ; s(100) = s(Σs(Σs(37))) = 117
    Σs(194) = Σs(496) = s(496) = 496 = s(652) = s(s(608)) = s(s(Σs(607)))
    Σs(607) = 608
    158 = Σs(157) = σ(157) = Σs(314/2) = 10+13+18+22+37+58
    Σs(158) = 418 = 836 / 2
    σ(158) = 240 = σ(209) ; σ(240) = 744

    28 in divisor sum context
    6 = σ(5) = 6
    12 = σ(6) = σ(11)
    28 = σ(12) = s(28)
    56 = σ(28) = σ(39) ; 39 = σ(18)
    120 = σ(56) = σ(95) ; 95 = 37+58 = 67+28
    360 = σ(120) = σ(323) = σ(290)-σ(145) ; σ(145) = 180 ; σ(290) = 540

  203. Paul Vaughan says:

    “However there are coincidences as yet unexplained, one of which concerns the number 163.”

    Demystification

    28 isn’t Heegner
    but is 744-level
    (along with top 4 Heegner nos.)

    ±28 +28 -28
    19 47 -9
    43 71 15
    67 95 39
    163 191 135

    47 = ΣΦ(29) = ΣΦ(58) = Σφ(145) = Σφ(323)
    71 = ΣΦ(41) = ΣΦ(55) = ΣΦ(1^2+2^2+3^2+4^2+5^2)
    95 = 37+58
    191 = ΣΦ(145) = ΣΦ(290) = Σφ(158) = ΣΦ(232)
    σ(135) = 240 ; Σδ(553) = 169
    ΣΣφ(135) = 104
    2 * 28 = σ(28) = 56 = σ(39) ; 39 = Σδ(63) ; 63+95 = 158
    ΣΣφ(9) = 8
    ΣΣφ(15) = 16
    ΣΣφ(28) = 32 = Σs(15) = Σs(3*5) = s(58) = Σφ(247)
    247 = average(-163,657) = average(Σφ(323),ΣΦ(323))
    = ΣΦ(490) ; 490 = p(19) ; 19 = x mod 24 for x=19,43,67,163
    900 = 657+ΣΦ(ΣΦ(ΣΦ(71))) = 657+ΣΦ(163)
    980 = 657+323

  204. Paul Vaughan says:

    012345

    (3+4)*(5-4)+2^(0+3) = 15
    (3+4)*(5-4)+2^(1+4) = 39
    (3+4)*(5-4)+2^(2+5) = 135

    (3+4)*(5+0)+2^(0+3) = 43
    (3+4)*(5+0)+2^(1+4) = 67
    (3+4)*(5+0)+2^(2+5) = 163

    (3+4)*(5+4)+2^(0+3) = 71
    (3+4)*(5+4)+2^(1+4) = 95
    (3+4)*(5+4)+2^(2+5) = 191

    “Numbers that are the sum of 4 but no fewer nonzero squares.”
    “Numbers of the form 4^i*(8*j+7), i >= 0, j >= 0.”
    https://oeis.org/A004215

    https://oeis.org/search?q=31+63+95+127+159+191

  205. Paul Vaughan says:

    Pell PPT 1
    13-10 = 3
    22-18 = 4
    18-13 = 5
    41 = 10+13+18 = Euler’s most lucky no.
    63 = 10+13+18+22
    100 = 10+13+18+22+37 ; Σδ(100) = 158
    158 = 10+13+18+22+37+58
    Pell PPT 2 linear combinations
    22 = 2*21 – 20
    37 = 5*20 – 3*21 = 100 – 63
    41 = 20+21
    58 = 2*29
    63 = 3*21
    100 = 5*20 ; Σδ(100) = 158
    158 = 5*20 + 2*29 = 100 + 58

    58 = perimeter difference
    58 = (20+21+29) – (3+4+5) = 70 – 12 = 2 * 29

    58 = (5+5)*(4+3)-(4*3)
    37 = (5*5)+(4*3)*(4-3)

  206. Paul Vaughan says:

    A000164 = A002635 – A025428

    Number of partitions of n into 4 squares.
    https://oeis.org/A002635

    Number of partitions of n into 4 nonzero squares.
    https://oeis.org/A025428

    Number of partitions of n into 3 squares (allowing part zero).
    https://oeis.org/A000164

    _ 7 15 23 28 31
    0 7 15 23 28 31
    1 39 47 55 60 63
    2 71 79 87 92 95
    3 103 111 119 124 127
    4 135 143 151 156 159
    5 167 175 183 188 191
    6 199 207 215 220 223
    7 231 239 247 252 255
    8 263 271 279 284 287
    9 295 303 311 316 319

    then 112+128k, 448+512k, 1792+2048k, etc. (A004215)

  207. Paul Vaughan says:

    reorganize
    _ 0 1 2 3 4 5
    0 7 28 112 448 1792 7168
    1 15 60 240 960 3840 15360
    2 23 92 368 1472 5888 23552
    3 31 124 496 1984 7936 31744
    4 39 156 624 2496 9984 39936
    5 47 188 752 3008 12032 48128
    6 55 220 880 3520 14080 56320
    7 63 252 1008 4032 16128 64512
    8 71 284 1136 4544 18176 72704
    9 79 316 1264 5056 20224 80896
    10 87 348 1392 5568 22272 89088
    https://oeis.org/search?q=7+15+23+31+39+47+55+63+71+79+87
    fib & luc pattern in errors is clear:
    https://en.wikipedia.org/wiki/Beatty_sequence#Examples

  208. Paul Vaughan says:

    1001 = 1673-672 = 1638+28+6+1-672
    1000 = 1673-672-1 = 1638+28+6-672
    378.378 = 0.378+378 = 378*(1001/1000) = 378*(1673-672)/(1673-672-1)

    32 = Σs(15) ; 15 = 5 * 3
    117 = average(-8190/140,8190/28) = 2*8190/140
    234 = 8190 / 28 – 8190 / 140 = 4 * 8190 / 140

    h(x) = harmonic mean of divisors of x

    h(x) x
    1 1
    2 6
    3 28
    5 140 ; 3,5 twin prime PPT: 4 = 140/(1+6+28) = √((h(140))^2-(h(28))^2)
    5 496
    6 270
    7 8128
    8 672
    9 1638
    10 6200
    11 2970
    15 8190

  209. Paul Vaughan says:

    generality:
    repeat binomial (pascal) converges to exponential in limit

    diversify exploration

    divisor sum
    8744 level
    σ(232) = 450
    σ(148) = 266
    σ(100) = 217
    σ(88) = 180
    σ(72) = 195

    repeat difference
    450
    266 184
    217 49 135
    180 37 12 123
    195 -15 52 -40 163

    163 = 1*450-4*266+6*217-4*180+1*195

    reverse order
    195
    180 15
    217 -37 52
    266 -49 12 40
    450 -184 135 -123 163

    hierarchical repeat difference
    (sum bottom row with reversed order or alternating sum of diagonal with unreversed order)
    21^2 = 441 = (1*450)-(1*450-1*266)+(1*450-2*266+1*217)-(1*450-3*266+3*217-1*180)+(1*450-4*266+6*217-4*180+1*195)

    analogous exploration (104 level)

    58
    37 21
    22 15 6
    18 4 11 -5
    13 5 -1 12 -17
    10 3 2 -3 15 -32

    10
    13 -3
    18 -5 2
    22 -4 -1 3
    37 -15 11 -12 15
    58 -21 6 5 -17 32

    63 = 10+13+18+22
    63 = (1*58)-(1*58-1*37)+(1*58-2*37+1*22)-(1*58-3*37+3*22-1*18)+(1*58-4*37+6*22-4*18+1*13)-(1*58-5*37+10*22-10*18+5*13-1*10)

    euler’s lucky nos. pell PPT 2
    79 = 2+3+5+11+17+41 = 158 / 2 = 316 / 4
    58 = 17+41 (top 2)
    58 = 29 * 2
    37 = (17+41)-(2+3+5+11)
    21 = 2+3+5+11 (lowest 4)
    21 = 63/3 = 58-37

    2212 = 79 * s(28) (anomalistic JEV)
    378 = (10+2*13+18)+(13+2*18+22)+(18+2*22+37)+(22+2*37+58)
    378, 441, & 504 are all divisible by 63 (6, 7, 8*)
    21^2 = average(378,σ(220))

    random sampling of insight uniformly distributed over time

  210. Paul Vaughan says:

    euler’s lucky divisor sums
    σ(2) = 3
    σ(3) = 4
    σ(5) = 6
    σ(11) = 12
    σ(17) = 18
    σ(41) = 42

    odd: row sums (repeat differences built in)
    3=3
    3=4-1
    5=6-2+1
    7=12-6+4-3
    9=18-6+0+4-7
    11=42-24+18-18+22-29

    reverse order, exclude divisor sum, & note row sums = 24
    42: empty row
    18: 24=24
    12: 6+18=24
    6: 6+0+18=24
    4: 2+4-4+22=24
    3: 1+1+3-7+29=27

    σ(2)=3 mnemonic: 3-2=1, 27-3=24
    σ(2)=3 is the 1 exception in both cases
    so there’s enough to solve given either 42=σ(41) or
    29 = 1*42-5*18+10*12-10*6+5*4-1*3
    29 = 1*41-5*17+10*11-10*5+5*3-1*2

  211. Paul Vaughan says:

    378 / 2 = 189 =
    +(1*1)
    -(1*1-1*2)
    +(1*1-2*2+1*3)
    -(1*1-3*2+3*3-1*7)
    +(1*1-4*2+6*3-4*7+1*11)
    -(1*1-5*2+10*3-10*7+5*11-1*19)
    +(1*1-6*2+15*3-20*7+15*11-6*19+1*43)
    -(1*1-7*2+21*3-35*7+35*11-21*19+7*43-1*67)
    +(1*1-8*2+28*3-56*7+70*11-56*19+28*43-8*67+1*163)

    row sums (with repeat differences built in)
    1 = 1
    1 = 2-1
    2 = 3-1+0
    3 = 7-4+3-3
    4 = 11-4+0+3-6
    5 = 19-8+4-4+7-13
    18 = 43-24+16-12+8-1-12
    55 = 67-24+0+16-28+36-37+25
    216 = 163-96+72-72+88-116+152-189+214 = 3^3 + 4^3 + 5^3 = 6^3

    10 = 18-5-3 = 18-5-2-1 = 18-4-3-1
    13 = 18-5 = 18-3-2 = 18-4-1
    18
    22 = 18+4 = 18+3+1
    37 = 55-18
    58 = 55+3 = 55+2+1

    ordered divisor sums with repeat differences
    σ(10) = 18
    σ(13) = 14 4
    σ(18) = 39 -25 29
    σ(22) = 36 3 -28 57
    σ(37) = 38 -2 5 -33 90
    σ(58) = 90 -52 50 -45 12 78
    triangle sum:
    +18
    +14+4
    +39-25+29
    +36+3-28+57
    +38-2+5-33+90
    +90-52+50-45+12+78 = 378

  212. Paul Vaughan says:

    “Numbers of the form 4^i*(8*j+7), i >= 0, j >= 0.”
    https://oeis.org/A004215

    4^i*(8*j+7) =
    average(-316,324)^i*((324-316)*j+(323-316))

    exclusively:
    sum = 323 = 17*19 = 196883-196560 for σ(n) = n+1

    σ(1) = 1 1+1
    σ(2) = 3
    σ(3) = 4
    σ(7) = 8
    σ(11) = 12
    σ(19) = 20
    σ(43) = 44
    σ(67) = 68
    σ(163) = 164

    inclusively:
    sum = 324 = 18^2
    378 = 323 + 55

    316 = 163+67+43+19+11+7+3+2+1
    216 = 378-(163-67)-(67-43)-(43-19)-(19-11)-(11-7)-(7-3)-(3-2)-(2-1) = 378-Φ(163)
    Φ(163) = (163-67)+(67-43)+(43-19)+(19-11)+(11-7)+(7-3)+(3-2)+(2-1) = 162

    104-level repeat difference
    58
    37 21
    22 15 6
    18 4 11 -5
    13 5 -1 12 -17
    10 3 2 -3 15 -32

    sum:
    +58
    +37+21
    +22+15+6
    +18+4+11-5
    +13+5-1+12-17
    +10+3+2-3+15-32 = 194

  213. Paul Vaughan says:

    prime sort of algorithm
    1 = s(163) = s(67) = s(43) = s(19) = s(11) = s(7) = s(3) = s(2)
    1 = s(41) = s(17) = s(11) = s(5) = s(3) = s(2)
    0 = s(1) —– neither prime nor euler lucky

    41 = σ(163)/4
    17 = σ(67)/4
    11 = σ(43)/4
    5 = σ(19)/4
    3 = σ(11)/4
    2 = σ(7)/4
    1 = σ(3)/4

    320 = σ(163)+σ(67)+σ(43)+σ(19)+σ(11)+σ(7)+σ(3)
    320 = 164+68+44+20+12+8+4

    euler lucky (prime) only
    316 = σ(163)+σ(67)+σ(43)+σ(19)+σ(11)+σ(7)
    316 = 164+68+44+20+12+8

    316 = 163+67+43+19+11+7+3+2+1
    158 = 10+13+18+22+37+58 = 316 / 2

    σ(163) = 164
    σ(67) = 68
    σ(43) = 44
    σ(19) = 20
    σ(11) = 12
    σ(7) = 8
    σ(3) = 4
    σ(2) = 3
    σ(1) = 1

    prime Heegner divisor sums & their differences
    484 = 164+68+44+20+12+8+4+3 + (164-68)+(68-44)+(44-20)+(20-12)+(12-8)+(8-4)+(4-3)
    484 = 323 + 164-3
    484 = 320 + 164 = 28+19+43+67+163+σ(163) = 324 + 160
    22 = √484 = √(28+19+43+67+163+σ(163))
    18 = √324 = √(average(28,19+43+67+163)+σ(163))

    160 = 22^2-18^2 = average(28,19+43+67+163)
    320 = 28+19+43+67+163
    640 = 44^2-36^2 = 72+88+100+148+232

    189 = 1*164-7*68+21*44-35*20+35*12-21*8+7*4-1*3
    378 =(1*164-7*68+21*44-35*20+35*12-21*8+7*4-1*3)*2

  214. Paul Vaughan says:

    Heegner nos. for
    f(n) = σ(n)
    f(n) = n
    f(n) = Φ(n)

    378 = f(163)+f(1)
    +1*f(163)
    -8*f(67)
    +28*f(43)
    -56*f(19)
    +70*f(11)
    -56*f(7)
    +28*f(3)
    -8*f(2)
    +1*f(1)

    378 / 2 = 189 =
    -1*f(163)
    +7*f(67)
    -21*f(43)
    +35*f(19)
    -35*f(11)
    +21*f(7)
    -7*f(3)
    +1*f(2)

    378 / 2 = 189 =
    +1*f(163)
    -7*f(67)
    +22*f(43)
    -40*f(19)
    +46*f(11)
    -34*f(7)
    +16*f(3)
    -4*f(2)
    +1*f(1)

    216 =
    +1*f(1)
    -7*f(2)
    +22*f(3)
    -40*f(7)
    +46*f(11)
    -34*f(19)
    +16*f(43)
    -4*f(67)
    +1*f(163)

    coefficients: sum pascal’s triangle levels 1 to 9
    +1 = +0+0+0+0+0+0+0+0+1
    -7 = +0+0+0+0+0+0+0+1-8
    +22 = +0+0+0+0+0+0+1-7+28
    -40 = +0+0+0+0+0+1-6+21-56
    +46 = +0+0+0+0+1-5+15-35+70
    -34 = +0+0+0+1-4+10-20+35-56
    +16 = +0+0+1-3+6-10+15-21+28
    -4 = +0+1-2+3-4+5-6+7-8
    +1 = +1-1+1-1+1-1+1-1+1

    189 = 1*162-7*66+22*42-40*18+46*10-34*6+16*2-4*1+1*1
    189 = 1*163-7*67+22*43-40*19+46*11-34*7+16*3-4*2+1*1
    189 = 1*164-7*68+22*44-40*20+46*12-34*8+16*4-4*3+1*1

    216 = 1*1-7*1+22*2-40*6+46*10-34*18+16*42-4*66+1*162
    216 = 1*1-7*2+22*3-40*7+46*11-34*19+16*43-4*67+1*163
    216 = 1*1-7*3+22*4-40*8+46*12-34*20+16*44-4*68+1*164

    378 = 2*162-8*66+28*42-56*18+70*10-56*6+28*2-8*1+2*1
    378 = 2*163-8*67+28*43-56*19+70*11-56*7+28*3-8*2+2*1
    378 = 2*164-8*68+28*44-56*20+70*12-56*8+28*4-8*3+2*1

    378 = 2*1-8*1+28*2-56*6+70*10-56*18+28*42-8*66+2*162
    378 = 2*1-8*2+28*3-56*7+70*11-56*19+28*43-8*67+2*163
    378 = 2*1-8*3+28*4-56*8+70*12-56*20+28*44-8*68+2*164

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