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

0,1

Comments

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

Links

Table of n, a(n) for n=0..104.

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

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

Sequence in context: A245638 A176220 A197285 * A352672 A188147 A187286

Adjacent sequences: A357014 A357015 A357016 * A357018 A357019 A357020

Keyword

nonn,cons

Author

Amiram Eldar, Sep 09 2022

Status

approved