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

%I A374289 #11 Feb 16 2025 08:34:07
%S A374289 1,25,49,121,125,169,289,343,361,529,625,841,961,1225,1331,1369,1681,
%T A374289 1849,2197,2209,2401,2809,3025,3125,3481,3721,4225,4489,4913,5041,
%U A374289 5329,5929,6125,6241,6859,6889,7225,7921,8281,8575,9025,9409,10201,10609,11449,11881,12167
%N A374289 5-rough powerful numbers: numbers k coprime to 6 such that if a prime p divides k then p^2 also divides k.
%C A374289 This sequence is closed under multiplication.
%C A374289 The least term that is not a power of a prime (A000961) is a(14) = 5^2*7^2 = 1225.
%H A374289 Amiram Eldar, <a href="/A374289/b374289.txt">Table of n, a(n) for n = 1..10000</a>
%H A374289 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RoughNumber.html">Rough Number</a>.
%H A374289 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A374289 Sum_{n>=1} 1/a(n) = 4*zeta(2)*zeta(3)/(7*zeta(6)) = (4/7) * A082695 = 1.1106265353... .
%t A374289 powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Flatten @ Table[6*n + {-1, 1}, {n, 0, 2000}], powQ]
%o A374289 (PARI) is(k) = gcd(k, 6) == 1 && ispowerful(k);
%Y A374289 Intersection of A007310 and A001694.
%Y A374289 Intersection of A001651 and A062739.
%Y A374289 A374290 is a subsequence.
%Y A374289 Cf. A000961, A082695.
%K A374289 nonn,easy
%O A374289 1,2
%A A374289 _Amiram Eldar_, Jul 02 2024