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

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

Author

Artur Jasinski, Aug 01 2025

Keywords

• nonn
• cons
• changed

Comments

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.

Examples

			0.70407248732078447829629819997862445809258378...
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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

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

Extensions

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