A091925 -id:A091925 - cp's OEIS frontend

A376913 Decimal expansion of Product_{k=1..8} Gamma(k/3).

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

Offset: 1

Paolo Xausa, Oct 11 2024

			5.2386596251856584103292320999763662681359773992...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002194, A091925.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376859 (m = 4), A376911 (m = 5 and m = 6), A376912 (m = 7).

First[RealDigits[640*Pi^3/(2187*Sqrt[3]), 10, 100]]

Equals 640*Pi^3/(2187*sqrt(3)) = 640*A091925/(3^7*A002194) (cf. eq. 90 in Weisstein link).

A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.0207800444333631028232547399039818253534109375...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 52.

Cf. A002117, A002194, A016777, A016779, A091925, A233091, A377558, A377559, A377560.

RealDigits[2Pi^3/(81Sqrt[3])+13Zeta[3]/27,10,100][[1]]

Equals Sum_{k>=0} 1/(3*k + 1)^3 (see Finch).
Equals -psi''(1/3)/54 (see Shamos).
Equals hypergeom([1/3, 1/3, 1/3, 1], [4/3, 4/3, 4/3], 1). - R. J. Mathar, Jul 14 2025

A377558 Decimal expansion of Pi^3/64 + 7*zeta(3)/16.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.01037296826200719010420286858471867099445163674...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 40.

Cf. A002117, A016813, A016815, A091925, A233091, A377557, A377559, A377560.

RealDigits[Pi^3/64+7Zeta[3]/16,10,100][[1]]

Equals Sum_{k>=0} 1/(4*k + 1)^3 (see Finch).
Equals -psi''(1/4)/128 = -(psi''(1/8) + psi''(5/8))/1024 (see Shamos).
Equals hypergeom([1/4, 1/4, 1/4, 1], [5/4, 5/4, 5/4], 1). - R. J. Mathar, Jul 14 2025

A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.00368551534795269706323013702486057315272784359...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 28.

Cf. A002117, A002194, A016921, A016923, A091925, A233091, A377557, A377558, A377559.

RealDigits[Pi^3/(36*Sqrt[3])+91*Zeta[3]/216,10,100][[1]]

Equals Sum_{k>=0} 1/(6*k + 1)^3 (see Finch).

Equals -psi''(1/6)/432 (see Shamos).

A058285 Continued fraction for Pi^3.

Original entry on oeis.org

31, 159, 3, 7, 1, 13, 2, 1, 3, 1, 12, 2, 2, 4, 34, 2, 43, 3, 1, 3, 2, 1, 1, 5, 1, 1, 4, 1, 5, 4, 2, 4, 11, 3, 3, 1, 1, 2, 1, 7, 2, 1, 1, 3, 1, 12, 3, 1, 9, 2, 1, 8, 23, 1, 45, 1, 1, 2, 1, 23, 3, 2, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 16, 1, 15, 1, 2, 4, 1, 2, 1, 12, 8, 1, 8, 2, 1, 7, 2, 2, 4, 1, 11, 2, 23

Offset: 0

Robert G. Wilson v, Dec 07 2000

			31.00627668029982017547631... = 31 + 1/(159 + 1/(3 + 1/(7 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 22 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A091925 Decimal expansion. - Harry J. Smith, Jun 22 2009

Mathematica
```
ContinuedFraction[ Pi^3, 100]
```
PARI
```
contfrac(Pi^3)
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^3); for (n=0, 20000, write("b058285.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 22 2009

More terms from Jason Earls, Jul 24 2001

A096388 Decimal expansion of Pi^3 - e^3.

Original entry on oeis.org

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

Offset: 2

Mohammad K. Azarian, Aug 10 2004

			10.920739757112152434547785412519677...

Cf. A000796, A001113, A073244, A091925, A091933.

RealDigits[Pi^3-E^3,10,120][[1]] (* Harvey P. Dale, Dec 07 2014 *)

Equals A091925 - A091933. - Michel Marcus, Mar 11 2018

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

A276120 Decimal expansion of zeta(3)/Pi^3.

Original entry on oeis.org

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

Offset: 0

Martin Renner, Sep 10 2016

zeta(3) is sometimes called Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is irrational. Despite this it is not known if this constant divided by Pi^3 is irrational or even transcendental.

			0.03876817960291679894...

Cf. A000796, A002117, A027431, A091925.

RealDigits[Zeta[3]/Pi^3, 10, 100][[1]] (* Amiram Eldar, Jul 07 2021 *)

zeta(3)/Pi^3 \\ Michel Marcus, Jul 07 2021

Equals A002117/A091925.

Offset corrected by Rick L. Shepherd, Nov 03 2016

A306604 Number of perfect squares in the half-open interval [Pi^(n-1), Pi^n).

Original entry on oeis.org

0, 1, 2, 2, 4, 8, 14, 23, 43, 75, 134, 236, 419, 743, 1316, 2333, 4135, 7329, 12992, 23026, 40813, 72338, 128218, 227259, 402806, 713955, 1265453, 2242956, 3975538, 7046456, 12489518, 22137096, 39236979, 69545736, 123266607, 218484372, 387253468, 686388899

Offset: 0

Alois P. Heinz, Feb 27 2019

nonn

Inspired by A306486.

			a(4) = 4: in the interval [Pi^3, Pi^4) = [31.006..., 97.409...) = are four perfect squares: 36, 49, 64, 81.

Alois P. Heinz, Table of n, a(n) for n = 0..4024

Cf. A000290, A000796, A002161, A091925, A092425, A102475, A102477, A306486.

a:= n-> (f-> f(n)-f(n-1))(i-> ceil(Pi^(i/2))):
seq(a(n), n=0..42);

a(n) = ceiling(Pi^(n/2)) - ceiling(Pi^((n-1)/2)).
a(n) = A102477(n) - A102477(n-1).
Sum_{i=0..n} a(i) = A102475(n) for n > 0.
Lim_{n->oo} a(n+1)/a(n) = sqrt(Pi) = 1.7724538509... = A002161.

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 04 2023

			0.276427204245986573092639825616889946783740795...

John M. Campbell, Applications of a class of transformations of complex sequences, arXiv:2212.13305 [math.NT], 2022. See p. 6.

Cf. A000796, A001008, A002805, A002897, A016639, A016813, A063448, A068465, A068466, A091925, A203142, A203143.

Cf. A359532.

First[RealDigits[N[(Gamma[1/8]Gamma[3/8]/(Gamma[1/4]Gamma[3/4]))^2/(6Sqrt[2]Pi)-4Log[2]/Pi,100]]]

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8)/(Gamma(1/4)*Gamma(3/4)))^2/(6*sqrt(2)*Pi).

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8))^2/(12*sqrt(2)*Pi^3).

cp's OEIS Frontend

A376913 Decimal expansion of Product_{k=1..8} Gamma(k/3).

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A377557 Decimal expansion of 2*Pi^3/(81*sqrt(3)) + 13*zeta(3)/27.

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A377558 Decimal expansion of Pi^3/64 + 7*zeta(3)/16.

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A377560 Decimal expansion of Pi^3/(36*sqrt(3)) + 91*zeta(3)/216.

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A058285 Continued fraction for Pi^3.

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A096388 Decimal expansion of Pi^3 - e^3.

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

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A276120 Decimal expansion of zeta(3)/Pi^3.

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A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.

A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).