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Powerful numbers whose sum of powerful divisors (including 1) is even.
2

%I #7 Sep 16 2024 09:23:44

%S 9,25,36,49,72,81,100,121,144,169,196,200,225,288,289,324,361,392,400,

%T 441,484,529,576,625,648,675,676,729,784,800,841,900,961,968,1089,

%U 1125,1152,1156,1225,1296,1323,1352,1369,1444,1521,1568,1600,1681,1764,1800,1849,1936

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

%C 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 Powerful numbers that have at least one odd prime factor in their prime factorization that has an even exponent.

%C Equivalently, powerful numbers whose odd part (A000265) is not an exponentially odd number (A268335).

%H Amiram Eldar, <a href="/A376217/b376217.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F 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 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 (PARI) is(k) = {my(f = factor(k), i = 1 + !(k % 2)); ispowerful(f) && #select(x -> !(x%2), f[i..#f~,2]) > 0;}

%Y Intersection of A001694 and A376216.

%Y Cf. A000265, A005117, A065487, A082695, A183097, A268335.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Sep 16 2024