cp's OEIS Frontend

A382294 Decimal expansion of the asymptotic mean of the excess of the number of Fermi-Dirac factors of k over the number of distinct prime factors of k when k runs over the positive integers.

%I A382294 #7 Mar 21 2025 10:04:41
%S A382294 1,3,6,0,5,4,4,7,0,4,9,6,2,2,8,3,6,5,2,2,9,9,8,9,2,6,3,8,3,7,6,8,9,9,
%T A382294 7,6,1,6,5,8,2,4,6,9,0,8,3,7,8,3,9,7,1,0,3,6,8,9,3,4,2,7,8,7,1,5,6,1,
%U A382294 4,9,7,6,6,7,4,9,7,7,1,7,9,1,4,6,0,6,5,2,2,8,2,9,7,5,0,8,5,4,1,4,8,7,3,5,9
%N A382294 Decimal expansion of the asymptotic mean of the excess of the number of Fermi-Dirac factors of k over the number of distinct prime factors of k when k runs over the positive integers.
%C A382294 Analogous to Sum_{p prime} 1/(p*(p-1)) (A136141), which is the asymptotic mean of the excess of the number of prime factors over the number of distinct prime factors (A046660).
%F A382294 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A382290(k).
%F A382294 Equal Sum_{k>=3} A088705(k) * P(k), where P(s) is the prime zeta function.
%F A382294 Equals Sum_{p prime} f(1/p), where f(x) = -x + Sum_{k>=0} x^(2^k)/(1+x^(2^k)).
%e A382294 0.13605447049622836522998926383768997616582469083783...
%t A382294 s[n_] := Module[{c = CoefficientList[Series[-x + Sum[x^(2^k)/(1+x^(2^k)), {k, 0, n}],{x, 0, 2^n}], x]},Sum[c[[i]] * PrimeZetaP[i-1], {i, 3, Length[c]-2}]]; RealDigits[s[10], 10, 120][[1]]
%o A382294 (PARI) default(realprecision, 120); default(parisize, 10000000);
%o A382294 f(x, n) = -x + sum(k = 0, n, x^(2^k)/(1+x^(2^k)));
%o A382294 sumeulerrat(f(1/p, 8))
%Y A382294 Cf. A001221, A046660, A064547, A088705, A136141, A382290.
%K A382294 nonn,cons
%O A382294 0,2
%A A382294 _Amiram Eldar_, Mar 21 2025