A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.

1, 49, 121, 169, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3481, 3721, 4489, 4913, 5041, 5329, 5929, 6241, 6859, 6889, 7921, 8281, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14161, 14641, 16129, 16807, 17161, 17689, 18769

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(25) = 7^2*11^2 = 5929.

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 A007775 and A001694.

Intersection of A229829 and A062739.

Intersection of A047201 and A374289.

Cf. A000961, A082695.

Programs

Mathematica

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[20000], CoprimeQ[#, 30] && powQ[#] &]

PARI

is(k) = gcd(k, 30) == 1 && ispowerful(k);

Formula

Sum_{n>=1} 1/a(n) = 80*zeta(2)*zeta(3)/(147*zeta(6)) = (80/147) * A082695 = 1.05773955745... .

In general, the sum of reciprocals of the p-rough powerful numbers is (zeta(2)*zeta(3)/zeta(6)) * Product_{prime q < p} ((q-1)*q/(q^2-q+1)).