A271727 - OEIS

%I #24 Nov 17 2024 16:41:43

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

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

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

%N Decimal expansion of the density of exponentially 2^n-numbers (A138302).

%H Juan Arias-de-Reyna, <a href="/A271727/b271727.txt">Table of n, a(n) for n = 0..1000</a>

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

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1602.04244">A fast computation of density of exponentially S-numbers</a>, arXiv:1602.04244 [math.NT], 2016.

%F Equals Product_{prime p} f(1/p), where f(x) = 1-x^3+Sum_{n>=2}(x^(2^n)-x^(1+2^n)).

%e 0.87249717935391281355800771433253186691958393977733373765412...

%t $MaxExtraPrecision = m = 500; em = 10; f[x_] := Log[1 - x^3 + Sum[x^(2^e) - x^(1 + 2^e), {e, 2, 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]] (* _Amiram Eldar_, Sep 09 2022 *)

%Y Density of A138302.

%Y Cf. A271726 (Expansion of log(f(x))).

%K nonn,cons

%O 0,1

%A _Juan Arias-de-Reyna_, Apr 13 2016