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

%I #18 Jul 09 2024 02:21:53

%S 144,324,400,576,784,1296,1600,1728,1936,2025,2304,2500,2704,2916,

%T 3136,3600,3969,4624,5184,5625,5776,5832,6400,7056,7744,8000,8100,

%U 8464,9216,9604,9801,10000,10816,11664,12544,13456,13689,13824,14400,15376,15876,17424,18225

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

%C 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 Michael De Vlieger, <a href="/A366854/b366854.txt">Table of n, a(n) for n = 1..10000</a>

%F This sequence is A126706(i)^m, m > 1.

%F A131605 = union of {1}, A303606, and {a(n)}.

%F A286708 = union of A303606, {a(n)}, and A052486.

%F A001597 = union of {1}, A246547, A303606, and {a(n)}.

%F A001694 = union of A246547, A303606, {a(n)}, and A052486.

%e Let b(n) = A126706(n).

%e a(1) = b(1)^2 = 12^2 = 144. Since 144 = 2^4*3^2, it is both powerful and a perfect power.

%e a(2) = b(2)^2 = 18^2 = 324.

%e a(3) = b(3)^2 = 20^2 = 400.

%e a(8) = b(1)^3 = 12^3 = 1728, etc.

%t nn = 20000; i = 1; k = 2;

%t MapIndexed[Set[S[First[#2]], #1] &,

%t Select[Range@ Sqrt[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &] ];

%t Union@ Reap[

%t While[j = 2;

%t While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2,

%t k++; i++] ][[-1, 1]]

%Y Cf. A001597, A001694, A052486, A120944, A126706, A131605, A246547, A286708, A303606.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Jan 01 2024