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

Author

Amiram Eldar, Nov 09 2024

Keywords

• nonn
• easy

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.

Crossrefs

Intersection of A001694 and A229125.

Intersection of A000037 and A377821.

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

Programs

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 &]]

PARI

is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x%2), e) == 1);

Formula

Sum_{n>=1} 1/a(n) = zeta(2) * P(3) = A013661 * A085541 = 0.28747301899596333866... .