A387620 Perfect powers that are neither prime powers nor products of primorials.

100, 196, 225, 324, 400, 441, 484, 676, 784, 1000, 1089, 1156, 1225, 1444, 1521, 1600, 1764, 1936, 2025, 2116, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3249, 3364, 3375, 3844, 3969, 4225, 4356, 4624, 4761, 4900, 5476, 5625, 5776, 5832, 5929, 6084, 6400, 6724

Offset

1,1

Comments

Perfect powers k^m, m > 1, where k is composite and weak (i.e., in A106543) and is not a primorial.

Intersection of A131605 and A369361 = A131605 \ A025487.

Proper subset of A369636, which in turn is a proper subset of A367268.

Superset of A380456.

Links

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

Index entries for sequences related to powerful numbers.

Example

 n    a(n)
-----------------------------------
 1    100 = 10^2 = 2^2 *  5^2
 2    196 = 14^2 = 2^2 *  7^2
 3    225 = 15^2 = 3^2 *  5^2
 4    324 = 18^2 = 2^2 *  3^4
 5    400 = 20^2 = 2^4 *  5^2
 6    441 = 21^2 = 3^2 *  7^2
 7    484 = 22^2 = 2^2 * 11^2
 8    676 = 26^2 = 2^2 * 13^2
 9    784 = 28^2 = 2^4 *  7^2
10   1000 = 10^3 = 2^3 *  5^3
11   1089 = 33^2 = 3^2 * 11^2
17   1764 = 42^2 = 2^2 *  3^2 * 7^2

Mathematica

nn = 10000; mm = Sqrt[nn]; i = 1; k = 2; fQ[x_] := And[CompositeQ[x], GCD @@ # == 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, ReverseSort[#]] != x] &@ FactorInteger[x][[;; , -1]]; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@Sqrt[nn], fQ]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++]][[-1, 1]]

Crossrefs

Cf. A001597, A001694, A002110, A007916, A024619, A106543, A126706, A131605, A286708, A367268, A369361, A369636, A380456.

Sequence in context: A068805 A369417 A231088 * A380456 A365745 A104023

Adjacent sequences: A387617 A387618 A387619 * A387621 A387622 A387623

Keyword

nonn

Author

Michael De Vlieger, Oct 09 2025

Status

approved