Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #21 Jun 19 2020 03:19:41
%S 5,6,8,2,8,5,4,9,3,7,4,6,8,0,6,9,5,4,3,2,6,0,3,6,5,7,6,1,9,1,9,1,6,2,
%T 9,6,7,2,4,4,0,4,5,0,9,3,1,9,7,8,6,3,8,3,9,4,5,2,6,3,2,7,2,1,5,8,1,1,
%U 9,8,6,0,1,5,7,5,7,6,4,4,1,8,4,3,8,0,6,9,6,3,6,3,7,4,4,9,2,7,6,3,1,0,9,6,1,2
%N Decimal expansion of B, a constant appearing in an asymptotic formula related to the exponential divisor function sigma^(e).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.11 Abundant numbers density constant p. 126.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>
%F B = lim_{N->inf} (1/N^2) * Sum_{n=1..N} sigma^(e)(n), where sigma^(e)(n) is the sum of all exponential divisors of n.
%F B = (1/2) * Product_{p prime} (1 + 1/(p*(p^2 - 1)) - 1/(p^2 - 1) + (1 - 1/p)*Sum_{k>=2} p^k/(p^(2k)-1)).
%e 0.5682854937...
%t digits = 10; maxPi = 10^5;
%t B = (1/2)*Product[1 + 1/(p*(p^2-1)) - 1/(p^2-1) + (1-1/p)*((Log[-(1/p^2)] - Log[1/p^2] + QPolyGamma[0, -(Log[-(1/p^2)]/Log[p]), p] - QPolyGamma[0, -(Log[1/p^2]/Log[p]), p])/(2*Log[p])), {p,Prime[Range[maxPi]]}];
%t RealDigits[N[B] // Chop, 10, digits][[1]]
%t $MaxExtraPrecision = 2000; Do[m = 2000; Clear[f]; f[p_] := (1 + 1/(p*(p^2 - 1)) - 1/(p^2 - 1) + (1 - 1/p)*Sum[p^k/(p^(2 k) - 1), {k, 2, kmax}]); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2]/2 * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 112]]], {kmax, 100, 1000, 100}] (* _Vaclav Kotesovec_, Jun 19 2020 *)
%Y Cf. A072691 (value of the same limit sum with sigma(n) instead of sigma^(e)(n)).
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Jul 29 2016
%E More digits from _Robert G. Wilson v_, Feb 25 2019
%E More digits from _Vaclav Kotesovec_, Jun 19 2020