A357016 Decimal expansion of the asymptotic density of numbers whose exponents in their prime factorization are squares (A197680).
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, 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, 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
Offset: 0
Examples
0.64111516135935143144770618384424460415920894040925...
Links
- Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015-2016.
Programs
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Mathematica
$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]]
Formula
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).
Comments