cp's OEIS Frontend

A357017 Decimal expansion of the asymptotic density of odd numbers whose exponents in their prime factorization are squares.

%I A357017 #5 Sep 09 2022 04:18:47
%S A357017 4,0,9,7,9,7,4,4,6,7,1,3,3,1,9,7,0,7,5,1,0,9,2,2,9,5,6,5,2,8,4,4,0,4,
%T A357017 9,9,9,8,2,3,0,1,6,3,9,3,9,0,6,7,2,7,3,1,1,6,9,2,2,6,8,1,6,3,7,6,2,1,
%U A357017 9,8,3,5,0,3,1,1,5,9,5,7,3,6,2,7,8,6,0,9,3,3,9,0,2,0,1,8,0,5,3,6,9,4,1,4,5
%N A357017 Decimal expansion of the asymptotic density of odd numbers whose exponents in their prime factorization are squares.
%C A357017 Equivalently, the asymptotic density of numbers whose sum of their exponential divisors (A051377) is odd (A357014).
%H A357017 Vladimir Shevelev, <a href="http://arxiv.org/abs/1510.05914">Exponentially S-numbers</a>, arXiv:1510.05914 [math.NT], 2015-2016.
%F A357017 Equals (1/2) * Product_{p odd prime} (1 + Sum_{k>=2} (c(k)-c(k-1))/p^k), where c(k) is the characteristic function of the squares (A010052).
%e A357017 0.40979744671331970751092295652844049998230163939067...
%t A357017 $MaxExtraPrecision = m = 1000; em = 100; f[x_] := Log[1 + Sum[x^(e^2), {e, 2, em}] - Sum[x^(e^2 + 1), {e, 1, em}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]
%Y A357017 Cf. A000290, A010052, A051377, A197680, A357014, A357016.
%K A357017 nonn,cons
%O A357017 0,1
%A A357017 _Amiram Eldar_, Sep 09 2022