A374289 5-rough powerful numbers: numbers k coprime to 6 such that if a prime p divides k then p^2 also divides k.

1, 25, 49, 121, 125, 169, 289, 343, 361, 529, 625, 841, 961, 1225, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3025, 3125, 3481, 3721, 4225, 4489, 4913, 5041, 5329, 5929, 6125, 6241, 6859, 6889, 7225, 7921, 8281, 8575, 9025, 9409, 10201, 10609, 11449, 11881, 12167

Offset: 1

Author

Amiram Eldar, Jul 02 2024

Keywords

• nonn
• easy

Comments

This sequence is closed under multiplication.

The least term that is not a power of a prime (A000961) is a(14) = 5^2*7^2 = 1225.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Rough Number.
• Index entries for sequences related to powerful numbers.

Crossrefs

Intersection of A007310 and A001694.

Intersection of A001651 and A062739.

A374290 is a subsequence.

Cf. A000961, A082695.

Programs

Mathematica

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Flatten @ Table[6*n + {-1, 1}, {n, 0, 2000}], powQ]

PARI

is(k) = gcd(k, 6) == 1 && ispowerful(k);

Formula

Sum_{n>=1} 1/a(n) = 4*zeta(2)*zeta(3)/(7*zeta(6)) = (4/7) * A082695 = 1.1106265353... .