A092732 -id:A092732 - cp's OEIS frontend

A376742 Decimal expansion of Product_{p prime} (p^3 + 1)/(p^3 - 1).

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

Offset: 1

Stefano Spezia, Oct 03 2024

			1.420308303489193353248184427065490...

E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986. See equation 1.2.8 at p. 5.

Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 408-409.

Cf. A000578, A001221, A002117, A013664, A034444, A092732, A183030.

RealDigits[Zeta[3]^2/Zeta[6],10,100][[1]]

PARI
```
prodeulerrat((p^3 + 1)/(p^3 - 1))
```

Equals zeta(3)^2/zeta(6) = Sum_{k>=1} 2^omega(k)/k^3. See Titchmarsh and Shamos.
Equals 945*zeta(3)^2/Pi^6.
Equals A157289 / A088453 = A013664 / A347328^2. - R. J. Mathar, Jul 14 2025

A257136 Decimal expansion of 2*Pi^6/945.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 16 2015

			2.034686123968898279429035859581841055803634980065707123684817328...

L. J. P. Kilford, Modular Forms: A Classical and Computational Introduction, Imperial College Press, (2008) p. 15.

Eric Weisstein's MathWorld, Eisenstein Series.
Index entries for transcendental numbers

Cf. A013664, A092732, A257134.

RealDigits[2*Pi^6/945, 10, 102] // First

2*Pi^6/945 \\ Charles R Greathouse IV, Oct 01 2022

2*Pi^6/945 = 2*zeta(6) = G_6(infinity), where the function G_k(z) is the Eisenstein nonzero modular form of weight k.

A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Sep 01 2011

The absolute value of the integral {x=0..Pi/2} x^5*log(sin(x )) dx or (d^5/da^5 (integral {x=0..Pi/2} sin(ax)*log(sin(x )) dx)) at a=0. The absolute value of m=2 of (-1)^(m+1)*(sum {n=1..infinity} (limit {a -> 0} (d^(2m+1)/da^(2m+1) ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(pi/2)^2(m+1)*log(2)/2/(m+1).

			0.11757583407233248206...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 1.441.2

Cf. A173623, A173624, A193716, A193717, A196456.

RealDigits[ N[(4 Pi^6*Log[2]-90 Pi^4*Zeta[3]+1350 Pi^2*Zeta[5]-5715 Pi^2*Zeta[7])/1536,150]][[1]]

Equals (4*A092732*A002162-90*A092425*A002117+1350*A002388*A013663-5715*A013665)/1536.

cp's OEIS Frontend

A376742 Decimal expansion of Product_{p prime} (p^3 + 1)/(p^3 - 1).

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A257136 Decimal expansion of 2*Pi^6/945.

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A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.

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cp's OEIS Frontend

A376742 Decimal expansion of Product_{p prime} (p^3 + 1)/(p^3 - 1).

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A257136 Decimal expansion of 2*Pi^6/945.

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A194657 Decimal expansion of (4*Pi^6*log(2) - 90*Pi^4*zeta(3) + 1350*Pi^2*zeta(5) - 5715*zeta(7))/1536.

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A194657 Decimal expansion of (4Pi^6log(2) - 90Pi^4zeta(3) + 1350Pi^2zeta(5) - 5715*zeta(7))/1536.