A164109 -id:A164109 - cp's OEIS frontend

A197110 Decimal expansion of Pi^4/120.

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

Offset: 0

R. J. Mathar, Oct 10 2011

Decimal expansion of the double zeta function zeta(2,2). Not to be confused with the Hurwitz zeta function of two arguments or with the second derivative of the Riemann zeta function.

			0.8117424... = A164109/40 .

R. E. Crandall and J. P. Buhler, On the evaluation of Euler sums, Exper. Math. 3 (1994), 275.
Wikipedia, Multiple zeta function
Index entries for transcendental numbers

Cf. A092425, A164109.

Maple
```
evalf(Pi^4/120) ;
```

First[RealDigits[Pi^4/120,10,100]] (* Geoffrey Critzer, Nov 03 2013 *)

Pi^4/120 \\ Charles R Greathouse IV, Apr 17 2015

zetamult([2,2]) \\ Charles R Greathouse IV, Apr 17 2015

Equals Sum_{n >=2} Sum_{m=1..n-1} 1/(n*m)^2.

More terms from D. S. McNeil, Oct 10 2011

Definition simplified by R. J. Mathar, Feb 05 2013

A231535 Decimal expansion of Pi^4/15.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2013

Under proper scaling, the radiation distribution density function in terms of frequency is given by prl(x) = x^3/(exp(x)-1), the Planck's radiation law. This constant is the integral of prl(x) from 0 to infinity and leads to the total amount of electromagnetic radiation emitted by a body.

Also, in an 8-dimensional unit-radius hypersphere, equals one-fifth of its surface (A164109), and twice the integral of r^2 over its volume.

			6.4939394022668291490960221792470074166485057115123614460978572926647...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Ilham A. Aliev, Ayhan Dil, Tornheim-like series, harmonic numbers and zeta values, arXiv:2008.02488 [math.NT], 2020, p. 4.
Stanislav Sykora, Surface Integrals over n-Dimensional Spheres
Stanislav Sykora, Volume Integrals over n-Dimensional Ellipsoids
Wikipedia, Planck's Law
Ke Xiao, Dimensionless Constants and Blackbody Radiation Laws, Electronic Journal of Theoretical Physics, 8(2011), 379-388, Eq.6.
Index entries for transcendental numbers

Cf. A000796, A013662, A081819, A164109.

RealDigits[Pi^4/15, 10, 105][[1]] (* Bruno Berselli, Nov 12 2013 *)

PARI
```
Pi^4/15 \\ Michel Marcus, Aug 07 2020
```

6*prodeulerrat(p^4/(p^4-1)) \\ Hugo Pfoertner, Aug 07 2020

Equals 6*zeta(4), see A013662. - Bruno Berselli, Nov 12 2013
Equals Gamma(4)*zeta(4) = 2*3*Product_{prime p} (p^4/(p^4-1)). - Stanislav Sykora, Oct 20 2014
Equals Sum_{n, m >= 1} H(n+m)/(n*m*(n+m)) where H(n) is the n-th harmonic number. See Aliev and Dil. - Michel Marcus, Aug 07 2020
From Amiram Eldar, Aug 14 2020: (Start)
Equals Integral_{x=0..oo} x^3/(exp(x)-1) dx.
Equals Integral_{x=0..1} log(x)^3/(x-1) dx. (End)
Equals psi'''(1), the third derivative of the digamma function at 1. - R. J. Mathar, Aug 29 2023

A164108 Decimal expansion of Pi^4/24.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 8-dimensional unit sphere.

			4.0587121264167682181850138620293796354053160696952259038...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Cornel Ioan Vălean, Problem 11993, The American Mathematical Monthly, Vol. 124, No. 7 (2017), p. 659; An Integral Related to Euler Sums, Solution to Problem 11993 by Abdelhak Berkane, ibid., Vol. 126, No. 4 (2019), pp. 373-374.
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, n-sphere.
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164106.

RealDigits[Pi^4/24, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^4/24 \\ G. C. Greubel, Apr 09 2017
```

Equals A164109/8 = A092425/24 = A072691*A102753.

Pi^4/240 = -Integral_{x=0..1} log(1-x)*log(1+x)^2/x dx (Vălean, 2017). - Amiram Eldar, Mar 26 2022

A164104 Decimal expansion of 8*Pi^2/3.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 5-dimensional unit sphere.

			Equals 26.318945069571622983558642666336...

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

Cf. A019692, A019694, A164102, A091925, A164107, A164109.

RealDigits[8*Pi^2/3, 10, 50][[1]] (* G. C. Greubel, Sep 11 2017 *)

PARI
```
8*Pi^2/3 \\ G. C. Greubel, Sep 11 2017
```

Equals 5*A164103 = 10 * A019699 * A019692.

A-number in formula corrected by R. J. Mathar, Aug 12 2010

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

A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 259, 408, 512, 536, 481, 378, 264, 166, 94, 49, 24, 10, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the ceiling is A164083.
The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.
There are only 17 nonzero terms. - G. C. Greubel, Sep 10 2017

			Table of approximate real values before taking integer part.
========================
n (2*Pi)^n / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, 2nd ed., New York: Springer-Verlag, p. 9, 1993.
H. S. M. Coxeter, Regular Polytopes, 3rd ed., New York: Dover, 1973.
D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions, New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere.

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255.

A164081 := proc(n) (2*Pi)^n/(n-1)! ; floor(%) ; end: seq(A164081(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Floor[(2*Pi)^n/(n - 1)!], {n, 1, 100}] (* G. C. Greubel, Sep 10 2017 *)

for(n=1,100, print1(floor((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 10 2017

a(n) = floor( (2*Pi)^n/(n-1)! ).

Definition corrected by R. J. Mathar, Sep 09 2009

A164082 Rounded value of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

6, 39, 124, 260, 408, 513, 537, 482, 379, 264, 166, 95, 50, 24, 11, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The floor of this real sequence is A164081, the ceiling is A164083.
The surface area of the n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/ A072479.
There are 18 nonzero terms in this sequence. - G. C. Greubel, Sep 11 2017

			Table of approximate real values before rounding up or down:
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164083.

A164082 := proc(n) (2*Pi)^n/(n-1)! ; round(%) ; end: seq(A164082(n),n=1..80) ; # R. J. Mathar, Sep 09 2009

Table[Round[(2*Pi)^n/(n - 1)!], {n, 1, 20}] (* G. C. Greubel, Sep 11 2017 *)

for(n=1,20, print1(round((2*Pi)^n/(n-1)!), ", ")) \\ G. C. Greubel, Sep 11 2017

a(n) = round(((2*Pi)^n)/(n-1)!).

Definition corrected by R. J. Mathar, Sep 09 2009

A164083 Ceiling of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

Original entry on oeis.org

7, 40, 125, 260, 409, 513, 537, 482, 379, 265, 167, 95, 50, 25, 11, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jonathan Vos Post, Aug 09 2009

nonn

The rounded values of this real sequence is A164082, the floor is A164081.

The surface area of n-dimensional sphere of radius r is n*V_n*r^(n-1); see A072478/A072479.

			Table of approximate real values before rounding up.
========================
n ((2*pi)^n) / (n-1)!
1 6.28318531 = A019692
2 39.4784176 = 2*A164102
3 124.025107 = 4*A091925
4 259.757576 = 8*A164109
5 408.026246
6 512.740903
7 536.941018
8 481.957131
9 378.528246
10 264.262568
11 166.041068
12 94.8424365
13 49.6593836
14 24.00147
15 10.7718345
16 4.5120955
17 1.77189576
18 0.654891141
19 0.228600133
20 0.075596684
========================

Conway, J. H. and Sloane, N. J. A. Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, p. 9, 1993.
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Sommerville, D. M. Y. An Introduction to the Geometry of n Dimensions. New York: Dover, p. 136, 1958.

Eric W. Weisstein, Hypersphere,

Cf. A072345, A072346, A072478, A072479, A074457, A122510, A154255, A164081, A164082.

Table[Ceiling[(2Pi)^n/(n-1)!],{n,60}] (* Harvey P. Dale, Jul 30 2020 *)

a(n) = ceiling(((2*pi)^n)/(n-1)!).

Definition corrected - R. J. Mathar, Sep 09 2009

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))).

cp's OEIS Frontend

A197110 Decimal expansion of Pi^4/120.

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A231535 Decimal expansion of Pi^4/15.

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A164108 Decimal expansion of Pi^4/24.

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A164104 Decimal expansion of 8*Pi^2/3.

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

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A164081 Floor of 2^(n-1) times the surface area of the unit sphere in 2n-dimensional space.

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