A376217 Powerful numbers whose sum of powerful divisors (including 1) is even.

9, 25, 36, 49, 72, 81, 100, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 900, 961, 968, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1369, 1444, 1521, 1568, 1600, 1681, 1764, 1800, 1849, 1936

Offset: 1

Author

Amiram Eldar, Sep 16 2024

Keywords

• nonn
• easy

Comments

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.

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

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to powerful numbers.

Crossrefs

Intersection of A001694 and A376216.

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

Programs

Mathematica

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]

PARI

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

Formula

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... .