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A376217 Powerful numbers whose sum of powerful divisors (including 1) is even.

%I A376217 #7 Sep 16 2024 09:23:44
%S A376217 9,25,36,49,72,81,100,121,144,169,196,200,225,288,289,324,361,392,400,
%T A376217 441,484,529,576,625,648,675,676,729,784,800,841,900,961,968,1089,
%U A376217 1125,1152,1156,1225,1296,1323,1352,1369,1444,1521,1568,1600,1681,1764,1800,1849,1936
%N A376217 Powerful numbers whose sum of powerful divisors (including 1) is even.
%C A376217 The primitive terms of A376216: all the terms of A376216 are of the form k*m, where m is a term of this sequence and k is a squarefree number that is coprime to m.
%C A376217 Powerful numbers that have at least one odd prime factor in their prime factorization that has an even exponent.
%C A376217 Equivalently, powerful numbers whose odd part (A000265) is not an exponentially odd number (A268335).
%H A376217 Amiram Eldar, <a href="/A376217/b376217.txt">Table of n, a(n) for n = 1..10000</a>
%H A376217 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A376217 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (9/7) * Product_{p prime} (1 + 1/(p*(p^2-1))) = A082695 - (9/7) * A065487 = 0.36050781682112605291... .
%t A376217 q[n_] := Module[{f = FactorInteger[n], i = 2 - Mod[n, 2]}, AllTrue[f[[;;, 2]], # > 1 &] && AnyTrue[f[[i;;-1, 2]], EvenQ]]; Select[Range[2000], q]
%o A376217 (PARI) is(k) = {my(f = factor(k), i = 1 + !(k % 2)); ispowerful(f) && #select(x -> !(x%2), f[i..#f~,2]) > 0;}
%Y A376217 Intersection of A001694 and A376216.
%Y A376217 Cf. A000265, A005117, A065487, A082695, A183097, A268335.
%K A376217 nonn,easy
%O A376217 1,1
%A A376217 _Amiram Eldar_, Sep 16 2024