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A376217
Powerful numbers whose sum of powerful divisors (including 1) is even.
2
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
1,1
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).
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... .
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]
PROG
(PARI) is(k) = {my(f = factor(k), i = 1 + !(k % 2)); ispowerful(f) && #select(x -> !(x%2), f[i..#f~, 2]) > 0; }
CROSSREFS
Intersection of A001694 and A376216.
Sequence in context: A090333 A137190 A044070 * A372724 A348234 A226647
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 16 2024
STATUS
approved