cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A357016 Decimal expansion of the asymptotic density of numbers whose exponents in their prime factorization are squares (A197680).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Sep 09 2022

Keywords

Comments

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

Examples

			0.64111516135935143144770618384424460415920894040925...
		

Crossrefs

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[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).