A191622 Decimal expansion of the growth constant for the partial sums of maximal unitary squarefree divisors.
6, 4, 9, 6, 0, 6, 6, 9, 9, 3, 3, 7, 3, 4, 1, 1, 9, 4, 7, 3, 3, 9, 0, 4, 8, 8, 0, 4, 8, 0, 2, 1, 2, 1, 2, 6, 7, 0, 3, 8, 1, 0, 8, 9, 9, 3, 1, 9, 8, 8, 2, 8, 8, 3, 9, 1, 8, 3, 2, 1, 0, 3, 9, 2, 6, 1, 3, 2, 0, 7, 1, 0, 4, 2, 8, 9, 5, 5, 1, 4, 6, 2, 7, 2, 0, 3, 5, 3, 5, 1, 9, 3, 7, 2, 1, 1, 9, 8, 0, 0, 7, 2, 0, 3, 8, 5
Offset: 0
Examples
0.64960669933734119473390488048021212670381089931988288391832103926132071...
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.
- Steven R. Finch, Unitarism and Infinitarism, February 25, 2004, Section 0.4. [Cached copy, with permission of the author]
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant beta).
Programs
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Mathematica
$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-2, 0, 2, 0, -1}, {0, -2, 0, 2, -5}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Jun 19 2019 *) -
PARI
prodeulerrat(1 - (p^2+p-1)/(p^3*(p+1))) \\ Amiram Eldar, Mar 17 2021
Formula
Equals Product_{primes p=2,3,5,7,...} ( 1 - (p^2+p-1)/(p^3*(p+1)) ).
The constant d2 in the paper by Cloutier et al. such that Sum_{k=1..x} 1/A057521(x) = d2*x + O(x^(1/2)). - Amiram Eldar, Oct 01 2019
Extensions
More terms from Amiram Eldar, Jun 19 2019
More terms from Vaclav Kotesovec, Jun 13 2021
Comments