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A366854 Powers k^m such that k is neither squarefree nor prime powers, and m > 1.

%I A366854 #18 Jul 09 2024 02:21:53
%S A366854 144,324,400,576,784,1296,1600,1728,1936,2025,2304,2500,2704,2916,
%T A366854 3136,3600,3969,4624,5184,5625,5776,5832,6400,7056,7744,8000,8100,
%U A366854 8464,9216,9604,9801,10000,10816,11664,12544,13456,13689,13824,14400,15376,15876,17424,18225
%N A366854 Powers k^m such that k is neither squarefree nor prime powers, and m > 1.
%C A366854 Analogous to A303606 = { k^m : Omega(k) = omega(k) > 1, m > 1 }, i.e., squarefree composite k (in A120944) raised to m > 1. Proper subset of A131605, itself a proper subset of A286708, which is in turn a proper subset of A126706. This sequence does not intersect Achilles numbers A052486.
%H A366854 Michael De Vlieger, <a href="/A366854/b366854.txt">Table of n, a(n) for n = 1..10000</a>
%F A366854 This sequence is A126706(i)^m, m > 1.
%F A366854 A131605 = union of {1}, A303606, and {a(n)}.
%F A366854 A286708 = union of A303606, {a(n)}, and A052486.
%F A366854 A001597 = union of {1}, A246547, A303606, and {a(n)}.
%F A366854 A001694 = union of A246547, A303606, {a(n)}, and A052486.
%e A366854 Let b(n) = A126706(n).
%e A366854 a(1) = b(1)^2 = 12^2 = 144. Since 144 = 2^4*3^2, it is both powerful and a perfect power.
%e A366854 a(2) = b(2)^2 = 18^2 = 324.
%e A366854 a(3) = b(3)^2 = 20^2 = 400.
%e A366854 a(8) = b(1)^3 = 12^3 = 1728, etc.
%t A366854 nn = 20000; i = 1; k = 2;
%t A366854 MapIndexed[Set[S[First[#2]], #1] &,
%t A366854   Select[Range@ Sqrt[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &] ];
%t A366854 Union@ Reap[
%t A366854   While[j = 2;
%t A366854     While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2,
%t A366854     k++; i++] ][[-1, 1]]
%Y A366854 Cf. A001597, A001694, A052486, A120944, A126706, A131605, A246547, A286708, A303606.
%K A366854 nonn,easy
%O A366854 1,1
%A A366854 _Michael De Vlieger_, Jan 01 2024