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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

Cf. A000265, A005117, A005361, A065463, A072587, A183097, A268335, A335341, A376217 (subsequence).

Sequence in context: A307579 A319927 A034046 * A069562 A072502 A195268

Adjacent sequences: A376212 A376213 A376215 * A376217 A376218 A376219

Keyword

nonn,easy

Author

Amiram Eldar, Sep 15 2024

Status

approved