A342477 - OEIS

%I #8 Mar 14 2021 05:14:46

%S 1,1,2,1,1,1,3,2,1,1,1,2,1,1,3,1,5,2,1,1,1,2,6,1,3,1,2,1,1,7,1,2,1,3,

%T 1,1,5,2,1,1,1,2,3,1,1,1,2,1,6,1,1,2,3,10,1,1,5,2,1,1,1,3,11,2,1,7,1,

%U 1,2,1,1,3,1,2,1,1,6,5,1,2,1,3,13,1,1,2

%N The squarefree part of the powerful numbers: a(n) = A007913(A001694(n)).

%H Amiram Eldar, <a href="/A342477/b342477.txt">Table of n, a(n) for n = 1..10000</a>

%H Vlad Copil and Laureniu Panaitopol, <a href="https://www.jstor.org/stable/43679072">A sequence attached to powerful numbers</a>, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 3 (2007), pp. 249-258; <a href="https://ad-astra.ro/2012/01/23/a-sequence-attached-to-powerful-numbers/">alternative link</a>.

%F Theorems 1-4 by Copil and Panaitopol (2007):

%F Sum_{k=1..n} a(k) ~ (3/Pi^2)*zeta(4/3)*n^(3/2) + O(sqrt(n)*log(n)).

%F Sum_{k=1..n} a(k)^2 ~ n/3 + O(n^(5/6)).

%F Sum_{k=1..n} 1/a(k) ~ (zeta(5/2)/zeta(5))*sqrt(n) + O(log(n)).

%F Product_{k=1..n} a(k) ~ exp(c*sqrt(n) + O(n^(1/3)*log(n))), where c = Sum_{k>=1} f(A005117(k)), and f(x) = log(x)/x^(3/2).

%t f[p_, e_] := p^Mod[e, 2]; sqfp[n_] := Times @@ f @@@ FactorInteger[n]; powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; sqfp /@ Select[Range[1000], powQ]

%Y Cf. A001694, A005117, A007913.

%K nonn

%O 1,3

%A _Amiram Eldar_, Mar 13 2021