A275480 - 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