A334451 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^5).

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

Offset

1,4

Comments

In general, for s>0, Product_{k>=1} (1 + 1/A002145(k)^(2*s+1))/(1 - 1/A002145(k)^(2*s+1)) = (2*s)! * (2^(2*s + 2) - 2) * zeta(2*s+1) / (Pi^(2*s+1) * A000364(s)). - Dimitris Valianatos, May 01 2020

In general, for s>1, Product_{k>=1} (1 + 1/A002145(k)^s)/(1 - 1/A002145(k)^s) = 2^s * (2^s - 1) * zeta(s) / (zeta(s, 1/4) - zeta(s, 3/4)).

References

B. C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, p. 64-65.

Links

Table of n, a(n) for n=1..105.

Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants, Feb 18 1996, p. 7-8.

Formula

A334451 / A334452 = 1488*zeta(5)/(5*Pi^5).

A334449 * A334451 = 90720*zeta(5)/Pi^10.

Example

1.0041818167708566903887269765658569606315819506367432882834249768697794496439...

Crossrefs

Cf. A002145, A243381, A334426, A334447.

Sequence in context: A101512 A085994 A179836 * A040019 A240776 A019768

Adjacent sequences: A334448 A334449 A334450 * A334452 A334453 A334454

Keyword

nonn,cons

Author

Vaclav Kotesovec, Apr 30 2020

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020

Status

approved