A377819 Powerful numbers that have no more than one even exponent in their prime factorization.

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 72, 81, 108, 121, 125, 128, 169, 200, 216, 243, 256, 288, 289, 343, 361, 392, 432, 500, 512, 529, 625, 648, 675, 729, 800, 841, 864, 961, 968, 972, 1000, 1024, 1125, 1152, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1728, 1849, 1944, 2000

Offset: 1

Author

Amiram Eldar, Nov 09 2024

Keywords

• nonn
• easy

Comments

Powerful numbers k such that A350388(k) is either 1 or a prime power with an even positive exponent (A056798 \ {1}).

Links

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

Crossrefs

Disjoint union of A335988 and A377818.

Intersection of A001694 and the complement of A377817.

Cf. A056798, A065487, A350388.

Programs

Mathematica

With[{max = 2000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], _?EvenQ] <= 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) = Product_{p prime} (1 + 1/(p*(p^2-1))) * (1 + Sum_{p prime} (p/(p^3-p+1))) = 1.84528389659572754387... .