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

%I A374290 #15 Feb 16 2025 08:34:07
%S A374290 1,49,121,169,289,343,361,529,841,961,1331,1369,1681,1849,2197,2209,
%T A374290 2401,2809,3481,3721,4489,4913,5041,5329,5929,6241,6859,6889,7921,
%U A374290 8281,9409,10201,10609,11449,11881,12167,12769,14161,14641,16129,16807,17161,17689,18769
%N A374290 7-rough powerful numbers: numbers k coprime to 30 such that if a prime p divides k then p^2 also divides k.
%C A374290 This sequence is closed under multiplication.
%C A374290 The least term that is not a power of a prime (A000961) is a(25) = 7^2*11^2 = 5929.
%H A374290 Amiram Eldar, <a href="/A374290/b374290.txt">Table of n, a(n) for n = 1..10000</a>
%H A374290 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RoughNumber.html">Rough Number</a>.
%H A374290 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A374290 Sum_{n>=1} 1/a(n) = 80*zeta(2)*zeta(3)/(147*zeta(6)) = (80/147) * A082695 = 1.05773955745... .
%F A374290 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)).
%t A374290 powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[20000], CoprimeQ[#, 30] && powQ[#] &]
%o A374290 (PARI) is(k) = gcd(k, 30) == 1 && ispowerful(k);
%Y A374290 Intersection of A007775 and A001694.
%Y A374290 Intersection of A229829 and A062739.
%Y A374290 Intersection of A047201 and A374289.
%Y A374290 Cf. A000961, A082695.
%K A374290 nonn,easy
%O A374290 1,2
%A A374290 _Amiram Eldar_, Jul 02 2024