login
Decimal expansion of Product_{k>=1} (1 + 1/A002476(k)^3).
6

%I #21 Jun 27 2020 16:02:20

%S 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,

%T 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,

%U 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

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

%C 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)).

%C 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)).

%C 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)).

%C 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).

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

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

%e 1.0036025402212598967043239333321878591705394771...

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

%K nonn,cons

%O 1,4

%A _Vaclav Kotesovec_, May 02 2020

%E More digits from _Vaclav Kotesovec_, Jun 27 2020