A357016 - OEIS

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

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OFFSET

0,1

COMMENTS

Equivalently, the asymptotic density of numbers with an odd number of exponential divisors (A049419).

LINKS

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

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

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).

EXAMPLE

0.64111516135935143144770618384424460415920894040925...

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]]

CROSSREFS

Cf. A000290, A010052, A049419, A197680, A357017.

Sequence in context: A215832 A021160 A354887 * A060780 A199391 A106333

Adjacent sequences: A357013 A357014 A357015 * A357017 A357018 A357019

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Sep 09 2022

STATUS

approved