cp's OEIS Frontend

A376216 Numbers whose sum of powerful divisors (including 1) is even.

%I A376216 #10 Sep 16 2024 09:23:49
%S A376216 9,18,25,36,45,49,50,63,72,75,81,90,98,99,100,117,121,126,144,147,150,
%T A376216 153,162,169,171,175,180,196,198,200,207,225,234,242,245,252,261,275,
%U A376216 279,288,289,294,300,306,315,324,325,333,338,342,350,360,361,363,369,387,392,396,400
%N A376216 Numbers whose sum of powerful divisors (including 1) is even.
%C A376216 The sequence of numbers whose number of powerful divisors (including 1, A005361) is even is A072587, which is the sequence of numbers that are not exponentially odd (A268335).
%C A376216 The primitive terms of this sequence are the powerful terms (A376217). If m is a powerful term then k*m is a term of this sequence for all squarefree numbers k that are coprime to m.
%C A376216 Numbers that have at least one odd prime factor in their prime factorization that has an even exponent.
%C A376216 Numbers whose odd part (A000265) is not an exponentially odd number (A268335).
%C A376216 Also, numbers k such that A335341(k) is even.
%C A376216 The asymptotic density of this sequence is 1 - (6/5) * A065463 = 0.15466935880100128871... .
%H A376216 Amiram Eldar, <a href="/A376216/b376216.txt">Table of n, a(n) for n = 1..10000</a>
%t A376216 q[n_] := Module[{f = Select[FactorInteger[n], First[#] == 2 || Last[#] > 1 &], i = 2 - Mod[n, 2]}, Length[f] > 0 && AnyTrue[f[[i;;-1, 2]], EvenQ]]; Select[Range[400], q]
%o A376216 (PARI) is(k) = {my(f = factor(k), i = 1 + !(k % 2)); #select(x -> !(x%2), f[i..#f~,2]) > 0;}
%Y A376216 Subsequence of A013929.
%Y A376216 Cf. A000265, A005117, A005361, A065463, A072587, A183097, A268335, A335341, A376217 (subsequence).
%K A376216 nonn,easy
%O A376216 1,1
%A A376216 _Amiram Eldar_, Sep 15 2024