A271727 Decimal expansion of the density of exponentially 2^n-numbers (A138302).

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, 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, 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

Offset: 0

Author

Juan Arias-de-Reyna, Apr 13 2016

Keywords

• nonn
• cons

Examples

			0.87249717935391281355800771433253186691958393977733373765412...
		

Links

• Juan Arias-de-Reyna, Table of n, a(n) for n = 0..1000
• Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015-2016.
• Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

Crossrefs

Density of A138302.

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

Programs

Mathematica

$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 *)

Formula

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