A191622 - OEIS

%I #36 Jun 29 2023 09:24:49

%S 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,

%T 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,

%U 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

%N Decimal expansion of the growth constant for the partial sums of maximal unitary squarefree divisors.

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

%H Maurice-Étienne Cloutier, <a href="http://hdl.handle.net/20.500.11794/28374">Les parties k-puissante et k-libre d’un nombre</a>, Thèse de doctorat, Université Laval (2018).

%H Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Cloutier/cloutier2.html">On the powerful and squarefree parts of an integer</a>, Journal of Integer Sequences,  Vol. 17 (2014), Article 14.6.6.

%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004, Section 0.4. [Cached copy, with permission of the author]

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant beta).

%F Equals Product_{primes p=2,3,5,7,...} ( 1 - (p^2+p-1)/(p^3*(p+1)) ).

%F 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

%e 0.64960669933734119473390488048021212670381089931988288391832103926132071...

%t $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 *)

%o (PARI) prodeulerrat(1 - (p^2+p-1)/(p^3*(p+1))) \\ _Amiram Eldar_, Mar 17 2021

%Y Cf. A055231, A057521.

%K cons,nonn

%O 0,1

%A _R. J. Mathar_, Jun 09 2011

%E More terms from _Amiram Eldar_, Jun 19 2019

%E More terms from _Vaclav Kotesovec_, Jun 13 2021