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

0,1

Comments

The partial sums grow Sum_{n=1..N} A055231(n) = (this constant)*N^2/2 +O(N^(3/2)).

Links

Table of n, a(n) for n=0..105.

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).

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

Example

0.64960669933734119473390488048021212670381089931988288391832103926132071...

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 *)

Prog

(PARI) prodeulerrat(1 - (p^2+p-1)/(p^3*(p+1))) \\ Amiram Eldar, Mar 17 2021

Crossrefs

Cf. A055231, A057521.

Sequence in context: A153630 A113276 A086034 * A021943 A334445 A168198

Adjacent sequences: A191619 A191620 A191621 * A191623 A191624 A191625

Keyword

cons,nonn

Author

R. J. Mathar, Jun 09 2011

Extensions

More terms from Amiram Eldar, Jun 19 2019

More terms from Vaclav Kotesovec, Jun 13 2021

Status

approved