A334449 Decimal expansion of Product_{k>=1} (1 + 1/A002144(k)^5).

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

Offset

1,5

Comments

In general, for s>0, Product_{k>=1} (1 + 1/A002144(k)^(2*s+1))/(1 - 1/A002144(k)^(2*s+1)) = Pi^(2*s+1) * A000364(s) * zeta(2*s+1) / ((2^(2*s+2) + 2) * (2*s)! * zeta(4*s+2)). - Dimitris Valianatos, May 01 2020

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

A334449 / A334450 = 4725*zeta(5)/(16*Pi^5).

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

Example

1.0003234751480716386036864273399423692652465522027379804075071648599638113746...

Crossrefs

Cf. A002144, A243380, A334424, A334445.

Sequence in context: A132408 A091821 A086035 * A268289 A201907 A003559

Adjacent sequences: A334446 A334447 A334448 * A334450 A334451 A334452

Keyword

nonn,cons

Author

Vaclav Kotesovec, Apr 30 2020

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020

Status

approved