A345159 Product_{p primes, k>=1} ((p^(3*k + 3) - 1)/(p^(3*k + 3) - p^3))^(1/p^k).

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

Offset

1,3

Links

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

Ramanujan's Papers, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, Formula (20), constant c.

Formula

Equals lim_{n->infinity} (A345160(n) / (n!)^3)^(1/n).

Example

1.080023050247205358427916943691762321424008892237822698674347551375648...

Mathematica

$MaxExtraPrecision = 1000; m = 600; prod = 1; s = 3; Do[Clear[f]; f[p_] := ((p^((k + 1)*s) - 1)/(p^((k + 1)*s) - p^s))^(1/p^k); cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x, m + 1]]; prod *= f[2]*Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 110]]; Print[prod], {k, 1, 100}]

Crossrefs

Cf. A001158, A345160, A345144, A345158.

Sequence in context: A186980 A171918 A265117 * A279418 A157244 A154191

Adjacent sequences: A345156 A345157 A345158 * A345160 A345161 A345162

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jun 10 2021

Status

approved