A334447 Decimal expansion of Product_{k>=1} (1 + 1/A002145(k)^4).

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

Offset: 1

Author

Vaclav Kotesovec, Apr 30 2020

Keywords

• nonn
• cons

Comments

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)) = 1 / (2 * (-1)^s * PolyGamma(s-1, 1/4) / (2^s * (2^s - 1) * Gamma(s) * zeta(s)) - 1).

Examples

			1.01284973750365824105373880963011203968450421655386945092221...
		

References

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

Links

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

Crossrefs

Cf. A002145, A243381, A334426, A334451.

Formula

A334447 / A334448 = 1/(PolyGamma(3, 1/4)/(8*Pi^4) - 1).

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

Extensions

More digits from Vaclav Kotesovec, Jun 27 2020