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A342477 The squarefree part of the powerful numbers: a(n) = A007913(A001694(n)).

%I A342477 #8 Mar 14 2021 05:14:46
%S A342477 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 A342477 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 A342477 1,2,1,1,3,1,2,1,1,6,5,1,2,1,3,13,1,1,2
%N A342477 The squarefree part of the powerful numbers: a(n) = A007913(A001694(n)).
%H A342477 Amiram Eldar, <a href="/A342477/b342477.txt">Table of n, a(n) for n = 1..10000</a>
%H A342477 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 A342477 Theorems 1-4 by Copil and Panaitopol (2007):
%F A342477 Sum_{k=1..n} a(k) ~ (3/Pi^2)*zeta(4/3)*n^(3/2) + O(sqrt(n)*log(n)).
%F A342477 Sum_{k=1..n} a(k)^2 ~ n/3 + O(n^(5/6)).
%F A342477 Sum_{k=1..n} 1/a(k) ~ (zeta(5/2)/zeta(5))*sqrt(n) + O(log(n)).
%F A342477 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 A342477 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 A342477 Cf. A001694, A005117, A007913.
%K A342477 nonn
%O A342477 1,3
%A A342477 _Amiram Eldar_, Mar 13 2021