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

Author

Amiram Eldar, Sep 15 2024

Keywords

• nonn
• easy

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

Crossrefs

Subsequence of A013929.

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

Programs

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]

PARI

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