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%I #4 Sep 27 2019 19:45:36
%S 6,8,7,8,2,7,1,3,9,4,4,3,6,2,4,8,8,1,0,6,3,5,1,0,8,2,4,5,4,9,8,7,0,9,
%T 8,3,2,0,3,0,9,5,8,7,5,3,0,1,0,1,5,2,1,7,1,0,5,6,4,0,1,6,9,0,8,8,7,4,
%U 8,4,9,1,6,4,6,2,8,2,9,6,3,5,9,4,7,0,7
%N Decimal expansion of the asymptotic density of numbers whose number of divisors is a power of 2 (A036537).
%H Vladimir Shevelev, <a href="http://doi.org/10.4064/aa8395-5-2016">S-exponential numbers</a>, Acta Arithmetica, Vol. 175, No. 4 (2016), pp. 385-395, <a href="https://www.math.bgu.ac.il/~shevelev/S_exp_numb.pdf">alternative link</a>.
%F Equals Product_{p prime} (1 - 1/p) * (1 + Sum_{i>=1} 1/p^(2^i-1)).
%F Equals lim_{k->oo} A036538(k)/2^k.
%e 0.687827139443624881063510824549870983203095875301015...
%t $MaxExtraPrecision = 1000; m = 1000; em = 10; f[x_] := Log[(1 - x)*(1 + Sum[x^(2^e - 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, 100][[1]]
%Y Cf. A036537, A036538.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Sep 27 2019
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