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A357017
Decimal expansion of the asymptotic density of odd numbers whose exponents in their prime factorization are squares.
3
4, 0, 9, 7, 9, 7, 4, 4, 6, 7, 1, 3, 3, 1, 9, 7, 0, 7, 5, 1, 0, 9, 2, 2, 9, 5, 6, 5, 2, 8, 4, 4, 0, 4, 9, 9, 9, 8, 2, 3, 0, 1, 6, 3, 9, 3, 9, 0, 6, 7, 2, 7, 3, 1, 1, 6, 9, 2, 2, 6, 8, 1, 6, 3, 7, 6, 2, 1, 9, 8, 3, 5, 0, 3, 1, 1, 5, 9, 5, 7, 3, 6, 2, 7, 8, 6, 0, 9, 3, 3, 9, 0, 2, 0, 1, 8, 0, 5, 3, 6, 9, 4, 1, 4, 5
OFFSET
0,1
COMMENTS
Equivalently, the asymptotic density of numbers whose sum of their exponential divisors (A051377) is odd (A357014).
LINKS
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015-2016.
FORMULA
Equals (1/2) * Product_{p odd prime} (1 + Sum_{k>=2} (c(k)-c(k-1))/p^k), where c(k) is the characteristic function of the squares (A010052).
EXAMPLE
0.40979744671331970751092295652844049998230163939067...
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[(1/2) * Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 105][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Sep 09 2022
STATUS
approved