A345158 Product_{p primes, k>=1} ((p^(2*k + 2) - 1)/(p^(2*k + 2) - p^2))^(1/p^k).

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

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} (A247951(n) / (n!)^2)^(1/n).

Example

1.19060271642680719164374162595970365073938314790455329116674114399345...

Mathematica

$MaxExtraPrecision = 1000; m = 600; prod = 1; s = 2; 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, 150}]

Crossrefs

Cf. A001157, A247951, A345144, A345159.

Sequence in context: A093070 A010533 A173201 * A019740 A372270 A188738

Adjacent sequences: A345155 A345156 A345157 * A345159 A345160 A345161

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jun 10 2021

Status

approved