A377820 Powerful numbers that have a single odd exponent in their prime factorization.

8, 27, 32, 72, 108, 125, 128, 200, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2888, 3087, 3125, 3200, 3267, 3528, 3872, 3888, 4232, 4500, 4563, 4608, 4913, 5000

Offset

1,1

Comments

First differs from A370786 at n = 124: A370786(124) = 27000 = 2^3 * 3^3 * 5*3 is not a term of this sequence.

Powerful numbers k such that A350389(k) is 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) * P(3) = A013661 * A085541 = 0.28747301899596333866... .

Mathematica

With[{max = 5000}, 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, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x%2), e) == 1);

Crossrefs

Intersection of A001694 and A229125.

Intersection of A000037 and A377821.

Cf. A013661, A085541, A246551, A350389, A370786.

Sequence in context: A102834 A376171 A370786 * A116002 A339595 A376173

Adjacent sequences: A377817 A377818 A377819 * A377821 A377823 A377824

Keyword

nonn,easy

Author

Amiram Eldar, Nov 09 2024

Status

approved