A377821 Powerful numbers that have no more than one odd exponent in their prime factorization.

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152

Offset: 1

Author

Amiram Eldar, Nov 09 2024

Keywords

• nonn
• easy

Comments

Powerful numbers k such that A350389(k) is either 1 or a prime power with an odd exponent (A246551).

Links

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

Crossrefs

Disjoint union of A000290 \ {0} and A377820.

Cf. A013661, A085541, A246551, A350389.

Programs

Mathematica

With[{max = 1200}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # == 1 || Count[FactorInteger[#][[;; , 2]], _?OddQ] <= 1 &]]

PARI

is(k) = if(k == 1, 1, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x%2), e) <= 1);

Formula

Sum_{n>=1} 1/a(n) = zeta(2) * (1 + P(3)) = A013661 * (1 + A085541) = 1.93240708584418977513... .