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

1,2

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.

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... .

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 &]]

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 A335988 and A377818.

Intersection of A001694 and the complement of A377817.

Cf. A056798, A065487, A350388.

Sequence in context: A087244 A080062 A249125 * A306531 A363722 A088949

Adjacent sequences: A377816 A377817 A377818 * A377820 A377821 A377823

Keyword

nonn,easy

Author

Amiram Eldar, Nov 09 2024

Status

approved