A376742 -id:A376742 - cp's OEIS frontend

A376845 Decimal expansion of Product_{p prime} (p^5 + 1)/(p^5 - 1).

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

Offset: 1

Stefano Spezia, Oct 06 2024

			1.0741508456720383646975062147556164766645757325...

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 p. 111.

Cf. A013663, A013668, A376742.

RealDigits[Zeta[5]^2/Zeta[10],10,100][[1]]

PARI
```
prodeulerrat((p^5 + 1)/(p^5 - 1))
```

Equals zeta(5)^2/zeta(10) = Sum_{k>=1} 2^omega(k)/k^5. See Titchmarsh and Shamos.

A385809 Decimal expansion of the Product_{p prime} (p^3-1)/(p^3+1).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Aug 01 2025

Product_{p prime} (p^(2*n)-1)/(p^(2*n)+1) are rational numbers A114362(n)/A114363(n) = zeta(4*n)/zeta(2*n)^2.

Product_{p prime} (p^(2*n+1)-1)/(p^(2*n+1)+1) = zeta(2*(2*n+1))/zeta(2*n+1)^2.

			0.70407248732078447829629819997862445809258378...

Richard Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011, Table 1 p. 2.

Cf. A002117, A013664, A112407, A114362, A114363, A376742.

RealDigits[Zeta[6]/Zeta[3]^2,10,105][[1]]

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

Equals zeta(6)/zeta(3)^2.

Equals 1 / A376742. - Amiram Eldar, Aug 01 2025

a(109) corrected by Georg Fischer, Aug 31 2025

cp's OEIS Frontend

A376845 Decimal expansion of Product_{p prime} (p^5 + 1)/(p^5 - 1).

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