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A328013
Decimal expansion of the growth constant for the partial sums of powerful part of n (A057521).
2
3, 5, 1, 9, 5, 5, 5, 0, 5, 8, 4, 1, 7, 1, 0, 6, 6, 4, 7, 1, 9, 7, 5, 2, 9, 4, 0, 3, 6, 9, 8, 5, 7, 8, 1, 7, 1, 8, 6, 0, 3, 9, 8, 0, 8, 2, 2, 5, 4, 0, 7, 8, 1, 4, 7, 1, 1, 4, 6, 4, 0, 3, 1, 4, 5, 4, 1, 7, 8, 3, 9, 8, 4, 7, 9, 7, 3, 5, 4, 0, 8, 9, 7, 7, 1, 3, 5, 8, 0, 3, 7, 5, 3, 6, 4, 6, 1, 6, 2, 0, 1, 1, 4, 5, 5
OFFSET
1,1
LINKS
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d’un nombre, Thèse de doctorat, Université Laval (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
FORMULA
The constant d1 in the paper by Cloutier et al. such that Sum_{k=1..x} A057521(k) = (d1/3)*x^(3/2) + O(x^(4/3)).
Sum_{k=1..x} 1/A055231(k) = d1*x^(1/2) + O(x^(1/3)).
Equals Product_{primes p} (1 + 2/p^(3/2) - 1/p^(5/2)).
EXAMPLE
3.51955505841710664719752940369857817186039808225407...
MATHEMATICA
$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 0, -2, 0, 1}, {0, 0, 6, 0, -5}, m]; RealDigits[(1 + 2/2^(3/2) - 1/2^(5/2))*(1 + 2/3^(3/2) - 1/3^(5/2))* Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 3, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
PROG
(PARI) prodeulerrat(1 + 2/p^3 - 1/p^5, 1/2) \\ Amiram Eldar, Jun 29 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 01 2019
EXTENSIONS
More terms from Vaclav Kotesovec, May 29 2020
STATUS
approved