A357017 Decimal expansion of the asymptotic density of odd numbers whose exponents in their prime factorization are squares.

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

Author

Amiram Eldar, Sep 09 2022

Keywords

• nonn
• cons

Comments

Equivalently, the asymptotic density of numbers whose sum of their exponential divisors (A051377) is odd (A357014).

Examples

			0.40979744671331970751092295652844049998230163939067...
		

Links

• Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015-2016.

Crossrefs

Cf. A000290, A010052, A051377, A197680, A357014, A357016.

Programs

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]]

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).