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

1,2

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.

Formula

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

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 &]]

Prog

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

Crossrefs

Disjoint union of A000290 \ {0} and A377820.

Cf. A013661, A085541, A246551, A350389.

Sequence in context: A348121 A080366 A001694 * A377846 A317102 A157985

Adjacent sequences: A377818 A377819 A377820 * A377823 A377824 A377825

Keyword

nonn,easy

Author

Amiram Eldar, Nov 09 2024

Status

approved