A390379 Powers k^m, 2 <= m <= 3, with k neither squarefree nor squareful.

144, 324, 400, 576, 784, 1600, 1728, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5625, 5776, 5832, 6400, 7056, 7744, 8000, 8100, 8464, 9216, 9604, 9801, 10816, 12544, 13456, 13689, 13824, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 21609, 21904

Offset

1,1

Comments

Perfect powers of numbers k that are neither squarefree nor squareful that are also not 4-full.

Intersection of A386762 and A391115 = A386762 \ A390381.

Union of A389947 and A392564, disjoint sets.

Distinct from A386762; A362762(42) = 20736 = 12^4 is not in this sequence.

Links

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

Index entries for sequences related to powerful numbers.

Example

Let s = A389947 and t = A392564.
Table of n, a(n) for select n:
 n    a(n)
-------------------------------------------
 1    144 = s(1)  = 12^2 = 2^4 *  3^2
 2    324 = s(2)  = 18^2 = 2^2 *  3^4
 3    400 = s(3)  = 20^2 = 2^4 *  5^2
 4    576 = s(4)  = 24^2 = 2^6 *  3^2
 5    784 = s(5)  = 28^2 = 2^4 *  7^2
 6   1600 = s(6)  = 40^2 = 2^6 *  5^2
 7   1728 = t(1)  = 12^3 = 2^6 *  3^3
 8   1936 = s(7)  = 44^2 = 2^4 * 11^2
 9   2025 = s(8)  = 45^2 = 3^4 *  5^2
10   2304 = s(9)  = 48^2 = 2^8 *  3^2
15   3600 = s(10) = 60^2 = 2^4 *  3^2 * 5^2
20   5832 = t(2)  = 18^3 = 2^3 *  3^6

Mathematica

nn = 25000; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], 1 == Min[#] < Max[#] &@ FactorInteger[#][[All, -1]] &]]; Union@ Reap[While[j = 2; While[And[j < 4, S[i]^j < nn], Sow[S[i]^j]; j++]; j > 2, i++] ][[-1, 1]]

Crossrefs

Cf. A001597, A001694, A002760, A013929, A024619, A126706, A131605, A168363, A286708, A332785, A386762, A389949, A390381, A391115, A392564.

Sequence in context: A380857 A389864 A386762 * A392424 A389947 A392069

Adjacent sequences: A390376 A390377 A390378 * A390380 A390381 A390382

Keyword

nonn,easy

Author

Michael De Vlieger, Jan 30 2026

Status

approved