%I #5 Sep 09 2022 04:19:00

%S 6,4,1,1,1,5,1,6,1,3,5,9,3,5,1,4,3,1,4,4,7,7,0,6,1,8,3,8,4,4,2,4,4,6,

%T 0,4,1,5,9,2,0,8,9,4,0,4,0,9,2,5,7,4,6,5,2,6,8,5,5,6,0,9,4,1,0,5,3,3,

%U 0,7,2,3,9,3,8,3,2,0,4,0,9,7,3,4,5,4,2,1,1,8,4,6,7,4,0,0,6,9,3,5,6,3,6,3,5

%N Decimal expansion of the asymptotic density of numbers whose exponents in their prime factorization are squares (A197680).

%C Equivalently, the asymptotic density of numbers with an odd number of exponential divisors (A049419).

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1510.05914">Exponentially S-numbers</a>, arXiv:1510.05914 [math.NT], 2015-2016.

%F Equals Product_{p prime} (1 + Sum_{k>=2} (c(k)-c(k-1))/p^k), where c(k) is the characteristic function of the squares (A010052).

%e 0.64111516135935143144770618384424460415920894040925...

%t $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[Exp[NSum[Indexed[c, k]*PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]

%Y Cf. A000290, A010052, A049419, A197680, A357017.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Sep 09 2022