A109693 Decimal expansion of Sum_{k>=1} 1/sigma(k)^2.

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

Offset

1,2

Links

Table of n, a(n) for n=1..87.

Formula

Product_{p prime} Sum_{k>=0} 1/sigma(p^k)^2.

Example

1.3064565120...

Mathematica

$MaxExtraPrecision = m = 1000; f[p_, m_] := 1 + Sum[(p - 1)^2/(p^(k + 1) - 1)^2, {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x, m]], {x, 0, m}], x]]*Range[m]; RealDigits[f[2, Infinity] * Exp[NSum[Indexed[c, n]*((PrimeZetaP[n] - 1/2^n)/n), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Nov 14 2020 *)

Prog

(PARI) N=1000000000 prodeuler(p=2, N, sum(k=1, 200/log(p), if(k==1, 1., 1./((p^k-1)/(p-1))^2)))*(1+1/N/log(N))

Crossrefs

Cf. A000203 (sigma function), A072861.

Sequence in context: A133170 A062542 A360173 * A188858 A199610 A285871

Adjacent sequences: A109690 A109691 A109692 * A109694 A109695 A109696

Keyword

cons,nonn

Author

Franklin T. Adams-Watters, Aug 07 2005

Extensions

More terms from Amiram Eldar, Nov 14 2020

Status

approved