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
Examples
1.3064565120...
Programs
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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 *) -
PARI
my(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))
Formula
Product_{p prime} Sum_{k>=0} 1/sigma(p^k)^2.
Extensions
More terms from Amiram Eldar, Nov 14 2020