A345144 Product_{p primes, k>=1} ((p^(k+1) - 1)/(p^(k+1) - p))^(1/p^k).

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

Offset

1,2

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 exp(1) * lim_{n->infinity} (A066780(n)^(1/n)) / n.

Example

1.561596846931024164326967889144555644364737646822232169945866457...

Mathematica

$MaxExtraPrecision = 1000; m = 500; prod = 1; Do[Clear[f]; f[p_] := ((p^(k + 1) - 1)/(p^(k + 1) - p))^(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}], 100]]; Print[prod], {k, 1, 200}]

Crossrefs

Cf. A001088, A066843, A066780, A124175.

Sequence in context: A323738 A222133 A198728 * A238135 A195449 A020798

Adjacent sequences: A345141 A345142 A345143 * A345145 A345146 A345147

Keyword

nonn,cons

Author

Vaclav Kotesovec, Jun 09 2021

Status

approved