This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A109693 #36 Dec 31 2024 15:39:33
%S A109693 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,
%T A109693 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,
%U A109693 3,4,2,6,4,5,7,6,6,8,2,8,7,1,5,9,9,8,1
%N A109693 Decimal expansion of Sum_{k>=1} 1/sigma(k)^2.
%F A109693 Product_{p prime} Sum_{k>=0} 1/sigma(p^k)^2.
%e A109693 1.3064565120...
%t A109693 $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 *)
%o A109693 (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))
%Y A109693 Cf. A000203 (sigma function), A072861.
%K A109693 cons,nonn
%O A109693 1,2
%A A109693 _Franklin T. Adams-Watters_, Aug 07 2005
%E A109693 More terms from _Amiram Eldar_, Nov 14 2020