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

144, 324, 400, 576, 784, 1296, 1600, 1728, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 5832, 6400, 7056, 7744, 8000, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 13824, 14400, 15376, 15876, 17424, 18225

Offset: 1

Author

Michael De Vlieger, Jan 01 2024

Keywords

• nonn
• easy

Comments

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.

Examples

			Let b(n) = A126706(n).
a(1) = b(1)^2 = 12^2 = 144. Since 144 = 2^4*3^2, it is both powerful and a perfect power.
a(2) = b(2)^2 = 18^2 = 324.
a(3) = b(3)^2 = 20^2 = 400.
a(8) = b(1)^3 = 12^3 = 1728, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

nn = 20000; i = 1; k = 2;
MapIndexed[Set[S[First[#2]], #1] &,
  Select[Range@ Sqrt[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &] ];
Union@ Reap[
  While[j = 2;
    While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2,
    k++; i++] ][[-1, 1]]

Formula

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

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

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

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

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