A164104 -id:A164104 - cp's OEIS frontend

A164109 Decimal expansion of Pi^4/3.

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 8-dimensional unit sphere.

			32.4696970113341457454801108962350370832425285...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, Hypersphere.
Ce Xu and Jianqiang Zhao, Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers, arXiv:2203.04184 [math.NT], 2022.
Index entries for transcendental numbers.

Cf. A019692, A019694, A164102, A164104, A091925, A164107.

Cf. A001008, A002805.

RealDigits[Pi^4/3,10,120][[1]] (* Harvey P. Dale, Jul 31 2014 *)

PARI
```
Pi^4/3 \\ G. C. Greubel, Sep 11 2017
```

Equals 8*A164108 = A019670*A091925 = A092425/3.

Pi^4/30 = Sum_{k>=1} H(k)*H(k+1)/(k+1)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 19 2025

A164103 Decimal expansion of 8*Pi^2/15.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 5-dimensional unit sphere.

For all nonnegative integers n, let V_n be the volume of the n-dimensional unit sphere. If n != 5, then V_n < V_5, this constant (see A072345). - Rick L. Shepherd, Feb 23 2014

			Equals 5.2637890139143245967117285332672806...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere.
Index entries for transcendental numbers.

RealDigits[8*Pi^2/15, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
8*Pi^2/15 \\ G. C. Greubel, Apr 09 2017
```

Equals A164104/5 = 4*A164102/15.

A164107 Decimal expansion of 16*Pi^3/15.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 7-dimensional unit sphere.

			Equals 33.07336179231980818717473607157482154904030780...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Equals 7*A164106.

Cf. A019692, A019694, A164102, A164104, A091925, A164109.

RealDigits[16*Pi^3/15, 10, 50][[1]] (* G. C. Greubel, Sep 11 2017 *)

16*Pi^3/15 \\ G. C. Greubel, Sep 11 2017

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))).

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A164109 Decimal expansion of Pi^4/3.

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A164103 Decimal expansion of 8*Pi^2/15.

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A164107 Decimal expansion of 16*Pi^3/15.

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

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