A380456 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) < A006530(k), where omega = A001221.

100, 196, 225, 400, 441, 484, 676, 784, 1000, 1089, 1156, 1225, 1444, 1521, 1600, 1764, 1936, 2025, 2116, 2500, 2601, 2704, 2744, 3025, 3136, 3249, 3364, 3375, 3844, 3969, 4225, 4356, 4624, 4761, 4900, 5476, 5625, 5776, 5929, 6084, 6400, 6724, 7056, 7225, 7396

Offset: 1

Author

Michael De Vlieger, Jul 25 2025

Keywords

• nonn

Comments

Perfect powers k^m, m > 1, for composite k in A080259.

Examples

			Table of n, a(n) for n = 1..12:
 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    400 = 20^2 = 2^4 *  5^2
 5    441 = 21^2 = 3^2 *  7^2
 6    484 = 22^2 = 2^2 * 11^2
 7    676 = 26^2 = 2^2 * 13^2
 8    784 = 28^2 = 2^4 *  7^2
 9   1000 = 10^3 = 2^3 *  5^3
10   1089 = 33^2 = 3^2 * 11^2
11   1156 = 34^2 = 2^2 * 17^2
12   1225 = 35^2 = 5^2 *  7^2
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A001597, A001694, A002110, A006530, A053669, A055932, A080259, A126706, A131605, A286708, A369417, A380446.

Programs

Mathematica

a053669[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q] ], q]; nn = 2^13; Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[a053669[#1] < #2[[-1, 1]], GCD @@ #2[[;; , -1]] > 1] & @@ {#, FactorInteger[#]} &]

Formula

Intersection of A131605 and A080259 = A131605 \ A055932 = A131605 \ A380446.