Eliora Ben-Gurion - cp's OEIS frontend

A383141 Decimal expansion of the obliquity (in degrees) of a planet at which the annual instellations received by the poles and the equator are identical.

5, 3, 8, 9, 6, 2, 3, 5, 8, 6, 2, 9, 5, 4, 2, 8, 7, 3, 1, 0, 1, 1, 2, 4, 6, 9, 2, 4, 2, 0, 8, 2, 1, 0, 8, 4, 5, 7, 9, 2, 0, 9, 5, 8, 3, 7, 4, 4, 7, 8, 6, 1, 1, 2, 2, 2, 7, 5, 5, 8, 9, 3, 4, 3, 6, 4, 9, 5, 4, 0, 1, 9, 1, 1, 0, 0, 1, 4, 7, 8, 7, 7, 7, 9, 8, 2, 6, 4, 7, 9, 8, 5, 9, 7, 7, 7, 5, 2, 5, 5, 7, 3, 4, 3, 5, 2, 7, 0, 1, 8, 5, 7, 4, 0, 5, 6, 5

Offset: 2

Eliora Ben-Gurion, Apr 17 2025

For more information, references, programs see A381254, which is the main entry for this problem.

			53.896235862954287310112...

Cf. A381254 (in radians).

Equals A381254*180/Pi.

A381254 Decimal expansion of the obliquity (in radians) of a planet at which the annual instellations received by the poles and the equator are identical.

Original entry on oeis.org

9, 4, 0, 6, 6, 6, 7, 7, 0, 2, 3, 9, 9, 9, 9, 6, 6, 3, 2, 1, 5, 8, 8, 1, 8, 6, 7, 9, 9, 3, 8, 5, 7, 9, 0, 5, 3, 2, 8, 8, 2, 0, 5, 4, 7, 1, 7, 1, 6, 9, 0, 5, 6, 4, 6, 8, 5, 0, 5, 4, 7, 1, 2, 0, 1, 2, 7, 4, 6, 7, 1, 4, 1, 3, 7, 7, 7, 8, 8, 7, 0, 7, 3, 4, 3, 7, 6, 7, 0, 3, 2, 1, 6, 3, 0, 8, 0, 7, 2, 4, 3, 4, 4, 7

Offset: 0

Eliora Ben-Gurion, Apr 17 2025

For an obliquity x, the normalized annual instellation coefficient at the equator is e(x) = (EllipticE(sin(x)^2) + sqrt(cos(x)^2) * EllipticE(-tan(x)^2)) / Pi, and at the poles is p(x) = sin(x), and the present constant is x where e(x) = p(x).
These coefficients are obtained by integrating over the sine of solar altitude over the course of one planetary year.
If the obliquity of a planet is greater than this value (for example, Uranus), then the poles would receive more instellation per year than the equator, which would result in a climate that inverts typical perceptions of those latitudes and the polar regions would be hotter than equatorial ones, in some cases resulting in an "ice belt" planet. However, these seasonal means would be accompanied by intense seasonal variations, as opposed to purely "tropical" polar regions.

			0.9406667702399996632...

David Ferreira, John Marshall, Paul A. O’Gorman, and Sara Seager, Climate at high-obliquity Icarus, vol. 243, pp. 236-248, 2014.
John Marshall, Climate at high-obliquity
D. M. Williams, J. F. Kasting, and L. A. Frakes, Low-latitude glaciation and rapid changes in the Earth's obliquity explained by obliquity-oblateness feedback, Nature, 1998 Dec 3;396(6710):453-5.
Worldbuilding Pasta, Climate Explorations: Obliquity

Cf. A383141.

FindRoot[(EllipticE[Sin[x]^2] + Sqrt[Cos[x]^2] * EllipticE[-Tan[x]^2]) / Pi == Sin[x], {x, 0.94}, WorkingPrecision -> 100]

\\ definition of ellM as in Mathematica's EllipticE[m]
ellM(k) = intnum(t=0, Pi/2, sqrt(1-k*sin(t)^2));
solve (x=0.9, 0.95, (ellM(sin(x)^2) + sqrt(cos(x)^2)*ellM(-tan(x)^2))/Pi - sin(x)) \\ Hugo Pfoertner, Apr 26 2025

Equals A383141*Pi/180.

A358029 Decimal expansion of the ratio between step sizes of the diatonic and chromatic semitones produced by a circle of 12 perfect fifths in Pythagorean tuning.

Original entry on oeis.org

1, 2, 6, 0, 0, 1, 6, 7, 5, 2, 6, 7, 0, 8, 2, 4, 5, 3, 5, 9, 3, 1, 2, 7, 6, 1, 2, 2, 6, 0, 3, 9, 2, 4, 2, 3, 3, 7, 1, 8, 1, 1, 5, 5, 7, 9, 2, 3, 2, 7, 6, 7, 8, 3, 3, 4, 1, 0, 6, 5, 2, 0, 1, 6, 1, 6, 2, 0, 8, 7, 4, 8, 0, 0, 8, 3, 1, 2, 2, 7, 8, 4, 6, 8, 8, 1, 4

Offset: 1

Eliora Ben-Gurion, Oct 25 2022

Pythagorean tuning is a form of tuning produced by repeated stacking of the perfect fifth, which has the frequency ratio of 3:2. A circle of 12 perfect fifths is approximately equal to the tuning system predominantly in use in the world today. If the perfect fifth is stacked 12 times and the resulting sequence is octave-reduced, then this divides the octave into 5 chromatic semitones which are equal to 2187/2048 (A229948), and 7 diatonic semitones which are equal to 256/243 (A229943). Diatonic semitones are those which are derived from a circle of 7 perfect fifths, the diatonic scale, and 5 chromatic semitones are a byproduct of an addition of 5 more perfect fifths, that is, another rotation, to the scale.

			1.2600167526708245359312761226039242337181155792327678334106520161...

Wikipedia, Pythagorean tuning.
Wikipedia, Semitone, section Pythagorean tuning.
Index entries for sequences related to music.

Cf. A131071, A221363, A258054, A257811.

Cf. A229943, A229948.

RealDigits[(7*Log[3] - 11*Log[2])/(8*Log[2] - 5*Log[3]), 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

Equals log(2187/2048) / log(256/243).
Equals log(A229948) / log(A229943).
Equals (7*log(3) - 11*log(2))/(8*log(2) - 5*log(3)).

A333477 Decimal expansion of largest negative solution to x! = x, or Gamma(x+1)=x, negated.

Original entry on oeis.org

3, 1, 4, 3, 5, 8, 0, 8, 8, 8, 3, 4, 9, 9, 8, 0, 0, 5, 8, 6, 9, 4, 3, 5, 8, 7, 8, 1, 8, 2, 0, 2, 2, 7, 8, 9, 9, 5, 6, 6, 6, 6, 9, 6, 0, 4, 0, 6, 1, 5, 5, 7, 3, 4, 5, 7, 7, 8, 9, 2, 4, 9, 0, 2, 4, 2, 8, 6, 4, 2, 9, 5, 3, 5, 6, 3, 8, 2, 5, 8, 9, 0, 7, 3, 9, 3

Offset: 1

Eliora Ben-Gurion, Mar 27 2020

Fixed point for Gamma(x+1) closest to 1 and 2.

By a mathematical coincidence, the negated expansion of the number is within 0.0633%, or 1 part in 1580 from Pi. Likewise, this constant is 1 part in 1580 away from -Pi.

			x = -3.1435808883499800586943587818...

Eric Weisstein's World of Mathematics, Gamma Function.

RealDigits[x /. FindRoot[Gamma[x + 1] == x, {x, -3.1}, WorkingPrecision -> 100], 10, 100][[1]] (* Vaclav Kotesovec, Apr 18 2020 *)

solve(x=3.1,3.2,gamma(1-x)+x) \\ Charles R Greathouse IV, Apr 18 2020

A331122 Nearest integer to Integral_{x=0..n} x^x dx.

Original entry on oeis.org

0, 1, 3, 15, 114, 1242, 17129, 284714, 5526741, 122592633, 3057488913, 84665033543, 2576896828787, 85495426794698, 3070641026296061, 118685141706060740, 4911825483278949553, 216697390123422589527, 10151899714746097960699, 503310218588181014061292

Offset: 0

Eliora Ben-Gurion, Jan 10 2020

nonn

The antiderivative of x^x cannot be described in terms of elementary functions.

a(1)=1 is the rounding of A083648.

			Integral_{X=0..4} x^x dx = 114.119062..., so a(4) = 114.
Integral_{x=0..7} x^x dx = 284713.7347218579997..., so a(7) = 284714.

Eric Weisstein's World of Mathematics, Sophomore's Dream

Cf. A000312, A083648.

Table[Round[NIntegrate[x^x,{x,0,n},WorkingPrecision->100]],{n,0,20}] (* Harvey P. Dale, Nov 04 2020 *)

for(k=0,19,print1(round(intnum(x=0,k,x^x)),", ")) \\ Hugo Pfoertner, Jan 18 2020

A330380 Decimal expansion of the y-coordinate for the largest solution to e^x = Gamma(x+1).

Original entry on oeis.org

1, 9, 8, 4, 0, 6, 1, 3, 0, 3, 1, 1, 2, 7, 6, 7, 7, 6, 9, 1, 1, 5, 0, 4, 2, 7, 2, 6, 0, 1, 9, 4, 7, 4, 8, 1, 5, 6, 2, 4, 2, 3, 0, 3, 8, 2, 9, 8, 2, 5, 2, 8, 7, 6, 3, 0, 6, 6, 9, 6, 4, 9, 9, 2, 2, 5, 5, 8, 3, 4, 1, 7, 4, 3, 3, 8, 8, 7, 6, 1, 2, 7, 3, 4, 9, 9, 1, 3, 6, 1, 2, 4, 8, 3, 3, 9, 3, 1, 1, 2, 8, 5, 1, 0, 4, 0, 4, 4, 9, 2, 4, 4, 7, 2, 3

Offset: 3

Eliora Ben-Gurion, Dec 12 2019

This number is the y-coordinate of the point at which the factorial function, Gamma(x+1), begins to exceed the exponential function.

			x = 5.29031609311977071072...
y = 198.40406130311276776911...

Robert Munafo, Notable Properties of Specific Numbers.

Cf. A078335.

RealDigits[x /. FindRoot[Gamma[Log[x] + 1] == x, {x, 200}, WorkingPrecision -> 120], 10, 115][[1]] (* Amiram Eldar, May 31 2021 *)

\p200
exp(solve (x=5,6,exp(x)-gamma(x+1))) \\ Hugo Pfoertner, Dec 12 2019

Equals exp(A078335).

A327926 a(n) = 99^n.

Original entry on oeis.org

1, 99, 9801, 970299, 96059601, 9509900499, 941480149401, 93206534790699, 9227446944279201, 913517247483640899, 90438207500880449001, 8953382542587164451099, 886384871716129280658801, 87752102299896798785221299, 8687458127689783079736908601

Offset: 0

Eliora Ben-Gurion, Nov 09 2019

			a(4) = 96059601 = 100^4 - 4*100^3 + 6*100^2 - 4*100 + 1 = 100000000 - 4000000 + 60000 - 400 - 1: 100000000 -> 96000000 -> 96060000 -> 96059600 -> 96059601.
a(8) = 99^8 = 9227446944279201.

Index entries for linear recurrences with constant coefficients, signature (99).

Cf. A096884, A007318.

a(n) = 99^n.
From Elmo R. Oliveira, Aug 11 2024: (Start)
G.f.: 1/(1-99*x).
E.g.f.: exp(99*x).
a(n) = 99*a(n-1) for n > 0. (End)

A325629 Floor of number of n-dimensional degrees in an n-sphere.

Original entry on oeis.org

2, 360, 41252, 3712766, 283634468, 19145326633, 1170076174384, 65816784809141, 3447793362911604, 16969079580805447, 7901760333122072321, 350023289756266797348, 14816864219294689084225

Offset: 0

Eliora Ben-Gurion, Sep 07 2019

nonn

Only the 0th and 1st terms of this sequence are exact values of n-degrees in an n-sphere, by definition. The 0-sphere, being 2 disconnected points at the ends of a segment, is trivial.
The number of degrees, minutes, seconds in an n-sphere is designed to approximate the size of an n-cube, m^n units in size, as m becomes increasingly small, observed from the center of the sphere. This makes a degree Pi/180 of a radian, a square degree (Pi/180)^2 of a steradian, a cubic degree (Pi/180)^3 of a 3-radian, etc.
The sequence has a maximum value at n = 20626 with a value of 1.3610489172...*10^4479, too large to be written here. I conjecture that the peak value of the function analytically is somewhere near 64800/Pi = 20626.48062...
At n = 56058 the sequence has a value of 281 (actual number 281.4089), meaning the 56058-dimensional sphere has less than 360 degrees. At n = 56070, the function has a value of 0.6978855, turning the rest of the sequence into a string of zeros.
An "N-sphere" is located in an N+1-dimensional space, 1-sphere being a circle, 2-sphere being an ordinary sphere, and so on.
From Jon E. Schoenfield, Sep 07 2019: (Start)
The maximum value of the continuous function is 1.361052727810610001492173640278424460497...*10^4479 and it occurs at 20626.48061662940750570152124725484602696... which is close to 64800/Pi, but it's actually 64799.99997461521504462375443773494034381.../Pi. That numerator appears to be 64800 - z/64800 + (27/10) * z^2 / 64800^3 - ... where z = zeta(2) = Pi^2 / 6. (End)

			Number of cubic degrees in a 3-sphere:
Surface area of a 3-sphere: 2*Pi^((3+1)/2) / ((3+1)/2 - 1)! = 2*Pi^2 / (2-1)! = 2*Pi^2.
Cubic degrees: 2*Pi^2 * (180/Pi)^3 = 11664000 / Pi = 3712766.512...
Number of tesseractic degrees in a 4-sphere:
Surface area of a 4-sphere: 2*Pi^((4+1)/2) / Gamma(5/2) = 2*Pi^(5/2) / (3*Pi^(1/2)/4) = 8*Pi^2/3.
Tesseractic degrees: 8*Pi^2/3 * (180/Pi)^4 = 2799360000 / Pi^2 = 283634468.641...

Eric Weisstein's World of Mathematics, Solid Angle.
Wikipedia, N-sphere
Wikipedia, Particular values of the gamma function

Surface area of k-dimensional sphere for k=2..8: A019692, A019694, A164102, A164104, A091925, A164107, A164109.

Cf. A125560.

a(n) = floor((2*Pi^((n+1)/2)/((n+1)/2-1)!)/(Pi/180)^n).
a(n) = floor((2*Pi^((n+1)/2)/(Gamma((n+1)/2)))/(Pi/180)^n).
a(n) = floor(2^(2n+1)*45^n*Pi^((n+1)/2-n)/((n+1)/2-1)!).
a(n) = floor(2^(2n+1)*45^n*Pi^((n+1)/2-n)/(Gamma((n+1)/2))).

A325911 Screaming numbers in base 16: numbers whose hexadecimal representation is AAAAAAA...

Original entry on oeis.org

10, 170, 2730, 43690, 699050, 11184810, 178956970, 2863311530, 45812984490, 733007751850, 11728124029610, 187649984473770, 3002399751580330, 48038396025285290, 768614336404564650, 12297829382473034410, 196765270119568550570, 3148244321913096809130

Offset: 1

Eliora Ben-Gurion, Sep 08 2019

In any base b > 10, we may express ten as a digit by using the letter A.

			a(10) = 733007751850_10 = AAAAAAAAAA_16.

Colin Barker, Table of n, a(n) for n = 1..800
Eric Weisstein's World of Mathematics, Hexadecimal
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A131865, A001025, A228774.

10Accumulate[16^Range[0, 31]] (* Alonso del Arte, Sep 17 2019 *)
LinearRecurrence[{17,-16},{10,170},20] (* Harvey P. Dale, Apr 02 2023 *)

a(n)={10*(16^n-1)/15} \\ Andrew Howroyd, Sep 08 2019

Vec(10*x / ((1 - x)*(1 - 16*x)) + O(x^20)) \\ Colin Barker, Sep 16 2019

a = 10
while a:
    a = a*16+10
    print(a)

def a(n): return int("A"*n, 16)
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Jan 17 2022

a(n) = Sum_{i=0..n} 10*16^(i).
a(n) = A131865(n-1)*10.
a(n) = 10*(16^n-1)/15. - Andrew Howroyd, Sep 08 2019
From Colin Barker, Sep 16 2019: (Start)
G.f.: 10*x / ((1 - x)*(1 - 16*x)).
a(n) = 17*a(n-1) - 16*a(n-2) for n>2.
(End)
E.g.f.: (2/3)*exp(x)*(-1 + exp(15*x)). - Stefano Spezia, Sep 17 2019

A309037 Exponential Demlo sequence, like 12345...54321, but for powers of 2 instead.

Original entry on oeis.org

2, 242, 24842, 2496842, 249936842, 24998736842, 2499974736842, 249999494736842, 24999989894736842, 2499999797894736842, 249999995957894736842, 24999999919157894736842, 2499999998383157894736842

Offset: 1

Eliora Ben-Gurion, Jul 08 2019

Lim_{n->infinity} a(n)/10^(2n-1) = 0.25, thus the first digits converge toward 24999999999999999999999...
In other words, Sum_{i>=1} 2^n/10^n = Sum_{i>=1} 5^(-n) = 5/(1-5) = 5/4 = 1.25. Excluding the 1 at the beginning of the number gives 0.25. Dividing each term by 2 gives the previous term with 1s attached on each side.
For example, 24998736842 / 2 = 12499368421.
In the set of {a(n)}, the final digits of a(n) eventually tend to be the repeating portion of 1/19 as n approaches infinity: ... 052631578947368421 05263157894736842.
If 8421... is analytically continued, 052631578947436... is obtained because Sum_{i>=1} 1/(2^n*10^n) is 1/19.
I propose that the Demlo function should be generalized, so that the function A002477(A000079(n)) produces this sequence. As another example, A002477(A000040(n)) should produce 2, 232, 23532, 2357532, 235817532, 23582417532, etc.

			For n = 4:
  2000000    8 - 2 = 6
   400000
    80000
    16000    4 - 1 = 3
      800
       40
  +     2
  -------
  2496842
For n = 12:
2*10^(24-2) + 4*10^(24-3) + 8*10^(24-4) + ... + 4096*10^11 + ... + 8*10^2 + 4*10^1 + 2
  20000000000000000000000    24 - 2 = 22
   4000000000000000000000
    800000000000000000000
    160000000000000000000
     32000000000000000000
      6400000000000000000
      1280000000000000000
       256000000000000000
        51200000000000000
        10240000000000000
         2048000000000000
          409600000000000    12 - 1 = 11
           20480000000000
            1024000000000
              51200000000
               2560000000
                128000000
                  6400000
                   320000
                    16000
                      800
                       40
  +                     2
  -----------------------
  24999999919157894736842

Seiichi Manyama, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (121, -2120, 2000).

Cf. A002477, A000079. Numbers produced from A000079 using A002477 algorithm.

a(n) = 2^1*10^0 + 2^2*10^1 + ... + 2^(n-1)*10^(n-2) + 2^n*10^(n-1) + 2^(n-1)*10^n + 2^(n-2)*10^(n+1) + ... + 2^2*10^(2n-3) + 2^1*10^(2n-2).
Conjectures from Colin Barker, Jul 16 2019: (Start)
G.f.: 2*x*(1 - 10*x)*(1 + 10*x) / ((1 - x)*(1 - 20*x)*(1 - 100*x)).
a(n) = (-80 - 3*4^n*5^(1+n) + 19*100^n) / 760.
a(n) = 121*a(n-1) - 2120*a(n-2) + 2000*a(n-3) for n>3.
(End)

cp's OEIS Frontend

User: Eliora Ben-Gurion

A383141 Decimal expansion of the obliquity (in degrees) of a planet at which the annual instellations received by the poles and the equator are identical.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A381254 Decimal expansion of the obliquity (in radians) of a planet at which the annual instellations received by the poles and the equator are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A358029 Decimal expansion of the ratio between step sizes of the diatonic and chromatic semitones produced by a circle of 12 perfect fifths in Pythagorean tuning.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A333477 Decimal expansion of largest negative solution to x! = x, or Gamma(x+1)=x, negated.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

A331122 Nearest integer to Integral_{x=0..n} x^x dx.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A330380 Decimal expansion of the y-coordinate for the largest solution to e^x = Gamma(x+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A327926 a(n) = 99^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A325629 Floor of number of n-dimensional degrees in an n-sphere.

Views

Author

Keywords

Comments

Examples