A334477 Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).

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

Offset: 1

Author

Vaclav Kotesovec, May 02 2020

Keywords

• nonn
• cons

Comments

In general, for s > 0, Product_{k>=1} (1 + 1/A002476(k)^(2*s+1)) / (1 - 1/A002476(k)^(2*s+1)) = sqrt(3) * (2*Pi)^(2*s + 1) * zeta(2*s + 1) * A002114(s) / ((2^(2*s + 1) + 1) * (3^(2*s + 1) + 1) * (2*s)! * zeta(4*s + 2)).

For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) / (1 - 1/A002476(k)^s) = (zeta(s, 1/6) - zeta(s, 5/6))*zeta(s) / ((2^s + 1)*(3^s + 1)*zeta(2*s)).

For s > 1, Product_{k>=1} (1 + 1/A002476(k)^s) * (1 + 1/A007528(k)^s) = 6^s * zeta(s) / ((2^s + 1) * (3^s + 1) * zeta(2*s)).

For s > 0, Product_{k>=1} ((A007528(k)^(2*s+1) - 1) / (A007528(k)^(2*s+1) + 1)) * ((A002476(k)^(2*s+1) + 1) / (A002476(k)^(2*s+1) - 1)) = 6 * A002114(s)^2 * (4*s + 2)! / ((2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2) = Bernoulli(2*s)^2 * (4*s + 2)! * (zeta(2*s + 1, 1/6) - zeta(2*s + 1, 5/6))^2 / (8*Pi^2 * (2^(4*s + 2) - 1) * (3^(4*s + 2) - 1) * Bernoulli(4*s + 2) * (2*s)!^2 * zeta(2*s)^2).

Examples

			1.0036025402212598967043239333321878591705394771...
		

Crossrefs

Cf. A002476, A175646, A334424, A334426, A334478, A334481.

Formula

A334477 / A334478 = 15*sqrt(3)*zeta(3)/Pi^3.

A334477 * A334479 = 810*zeta(3)/Pi^6.

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020