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

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, 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, 1, 2, 1, 1, 3, 1, 2, 1, 1, 6, 5, 1, 2, 1, 3, 13, 1, 1, 2

Offset: 1

Author

Amiram Eldar, Mar 13 2021

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vlad Copil and Laureniu Panaitopol, A sequence attached to powerful numbers, Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, Nouvelle Série, Vol. 50 (98), No. 3 (2007), pp. 249-258; alternative link.

Crossrefs

Cf. A001694, A005117, A007913.

Programs

Mathematica

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]

Formula

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

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

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

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

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).