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A376216
Numbers whose sum of powerful divisors (including 1) is even.
2
9, 18, 25, 36, 45, 49, 50, 63, 72, 75, 81, 90, 98, 99, 100, 117, 121, 126, 144, 147, 150, 153, 162, 169, 171, 175, 180, 196, 198, 200, 207, 225, 234, 242, 245, 252, 261, 275, 279, 288, 289, 294, 300, 306, 315, 324, 325, 333, 338, 342, 350, 360, 361, 363, 369, 387, 392, 396, 400
OFFSET
1,1
COMMENTS
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).
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.
Numbers that have at least one odd prime factor in their prime factorization that has an even exponent.
Numbers whose odd part (A000265) is not an exponentially odd number (A268335).
Also, numbers k such that A335341(k) is even.
The asymptotic density of this sequence is 1 - (6/5) * A065463 = 0.15466935880100128871... .
LINKS
MATHEMATICA
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]
PROG
(PARI) is(k) = {my(f = factor(k), i = 1 + !(k % 2)); #select(x -> !(x%2), f[i..#f~, 2]) > 0; }
CROSSREFS
Subsequence of A013929.
Sequence in context: A307579 A319927 A034046 * A069562 A072502 A195268
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 15 2024
STATUS
approved