cp's OEIS Frontend

A382295 Decimal expansion of the asymptotic mean of the number of ways to factor k into "Fermi-Dirac primes" when k runs over the positive integers.

%I A382295 #5 Mar 21 2025 10:04:52
%S A382295 1,7,8,7,6,3,6,8,0,0,1,6,9,4,4,5,6,6,6,9,8,8,6,3,2,9,3,9,4,8,9,4,5,9,
%T A382295 8,8,1,4,6,5,9,0,0,4,6,1,3,7,0,0,2,2,6,4,1,1,6,7,3,2,9,5,4,5,6,6,6,3,
%U A382295 7,5,1,3,9,5,4,3,4,0,2,5,1,5,5,1,5,5,0,8,8,3,3,3,5,8,7,1,3,7,5,6,1,5,6,0,4
%N A382295 Decimal expansion of the asymptotic mean of the number of ways to factor k into "Fermi-Dirac primes" when k runs over the positive integers.
%F A382295 Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A050377(k).
%F A382295 Equals Product_{p prime} f(1/p), where f(x) = (1-x) / Product_{k>=0} (1 - x^(2^k)).
%e A382295 1.78763680016944566698863293948945988146590046137002...
%t A382295 $MaxExtraPrecision = 1500; m = 1500; em = 50; f[x_] := Log[1-x] - Sum[Log[1-x^(2^k)], {k, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m]]; RealDigits[Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]
%o A382295 (PARI) default(realprecision, 120); default(parisize, 10000000);
%o A382295 f(x, n) = (1-x) / prod(k = 0, n, (1 - x^(2^k)));
%o A382295 prodeulerrat(f(1/p, 10))
%Y A382295 Cf. A005117 (positions of 1's in A050377), A050377, A082293 (positions of 2's), A330687 (positions of records).
%K A382295 nonn,cons
%O A382295 1,2
%A A382295 _Amiram Eldar_, Mar 21 2025