cp's OEIS Frontend

A380601 Decimal expansion of the asymptotic mean of the ratio A322483(k)/A000005(k).

%I A380601 #6 Jan 28 2025 01:48:44
%S A380601 8,5,9,8,0,6,7,7,9,3,3,0,3,4,3,6,3,3,1,1,2,4,4,7,6,7,5,9,4,9,4,1,8,3,
%T A380601 2,4,6,6,5,1,5,8,0,9,5,5,1,3,8,5,6,6,1,1,2,7,7,1,5,4,6,4,8,9,4,9,1,3,
%U A380601 4,3,3,0,8,5,8,7,6,9,4,9,7,3,4,2,3,7,6,4,8,4,8,5,9,3,5,3,5,2,4,5,4,4,8,4,5
%N A380601 Decimal expansion of the asymptotic mean of the ratio A322483(k)/A000005(k).
%F A380601 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A322483(k)/A000005(k).
%F A380601 Equals Product_{p prime} (1/2 - ((p-1)/4) * (3*log(1-1/p) + log(1+1/p))).
%e A380601 0.85980677933034363311244767594941832466515809551385...
%t A380601 $MaxExtraPrecision = 1000; m = 1000; f[x_] := (2*x + 3*(x-1)*Log[1 - x] + (x-1)*Log[1+x])/(4*x); c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]
%o A380601 (PARI) default(realprecision, 120); default(parisize, 30000000);
%o A380601 my(m = 1024, x = 'x + O('x^m), v); v = Vec((2*x + 3*(x-1)*log(1-x) + (x-1)*log(1+x))/(4*x)); prodeulerrat(sum(i=1, #v, v[i]/p^(i-1)))
%Y A380601 Cf. A000005, A322483, A380602 (mean of the inverse ratio).
%Y A380601 Similar constants: A308043, A361060, A361062.
%K A380601 nonn,cons
%O A380601 0,1
%A A380601 _Amiram Eldar_, Jan 27 2025