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Decimal expansion of Sum_{n>=1} 1/sigma_2(n).
2

%I #30 Sep 03 2024 09:23:24

%S 1,5,3,7,8,1,2,8,9,1,8,2,7,2,5,6,1,6,2,5,3,8,6,6,1,0,0,2,7,3,8,2,6,8,

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

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

%N Decimal expansion of Sum_{n>=1} 1/sigma_2(n).

%F Product_p Sum_{k>=0} 1/sigma_2(p^k).

%e 1.5378128918272561625386610027382683309193600494732235492961768965942633...

%t $MaxExtraPrecision = 1000; Do[Clear[f]; f[p_] := (1 + Sum[(p^2 - 1)/(p^(2*e + 2) - 1), {e, 1, emax}]); m = 1000; cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2]*Exp[N[Sum[cc[[n]] * (PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {emax, 100, 400, 100}] (* _Vaclav Kotesovec_, Sep 19 2020 *)

%o (PARI) N=10^9; prodeuler(p=2,N, sum(k=1,200/log(p),if(k==1,1.,1./((p^(2*k)-1)/(p^2-1))))) \\ The output is 1.537812891756...

%Y Cf. A001157 (sigma_2), A064602.

%K cons,nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Aug 07 2005

%E More digits from _Vaclav Kotesovec_, Sep 19 2020