A334445 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^4).

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

Offset

1,5

Comments

In general, for s>1, Product_{k>=1} (1 + 1/A002144(k)^s)/(1 - 1/A002144(k)^s) = (zeta(s, 1/4) - zeta(s, 3/4)) * zeta(s) / (2^s * (2^s + 1) * zeta(2*s)).

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

A334445 / A334446 = 35*(PolyGamma(3, 1/4)/(8*Pi^4) - 1)/34.

A334445 * A334447 = 1680 / (17*Pi^4).

Example

1.001649664033000425378578071929390888273984404386699300089837...

Crossrefs

Cf. A002144, A243380, A334424, A334449.

Sequence in context: A086034 A191622 A021943 * A168198 A177898 A082209

Adjacent sequences: A334442 A334443 A334444 * A334446 A334447 A334448

Keyword

nonn,cons

Author

Vaclav Kotesovec, Apr 30 2020

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020

Status

approved