A269404 Decimal expansion of Product_{k >= 1} (1 + 1/prime(k)^6).

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

Offset

1,4

Comments

More generally, Product_{k >= 1} (1 + 1/prime(k)^m) = zeta(m)/zeta(2*m), where zeta(m) is the Riemann zeta function.

Links

Table of n, a(n) for n=1..120.

Eric Weisstein's World of Mathematics, Prime Products.

Formula

Equals zeta(6)/zeta(12).

Equals 675675/(691*Pi^6).

Equals Sum_{k>=1} 1/A005117(k)^6 = 1 + Sum_{k>=1} 1/A113851(k). - Amiram Eldar, Jun 27 2020

Example

1.0170927691304992766432721330979099204922190794941...

Mathematica

RealDigits[Zeta[6]/Zeta[12], 10, 120][[1]]
RealDigits[675675/(691 Pi^6), 10, 120][[1]]

Prog

(PARI) zeta(6)/zeta(12) \\ Amiram Eldar, Jun 11 2023

Crossrefs

Cf. A005117, A082020, A113851, A157289, A157290, A157291.

Sequence in context: A378124 A234355 A021145 * A296433 A021589 A093444

Adjacent sequences: A269401 A269402 A269403 * A269405 A269406 A269407

Keyword

nonn,cons

Author

Ilya Gutkovskiy, Feb 25 2016

Status

approved