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

%I A380456 #35 Aug 07 2025 18:11:53
%S A380456 100,196,225,400,441,484,676,784,1000,1089,1156,1225,1444,1521,1600,
%T A380456 1764,1936,2025,2116,2500,2601,2704,2744,3025,3136,3249,3364,3375,
%U A380456 3844,3969,4225,4356,4624,4761,4900,5476,5625,5776,5929,6084,6400,6724,7056,7225,7396
%N A380456 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) < A006530(k), where omega = A001221.
%C A380456 Perfect powers k^m, m > 1, for composite k in A080259.
%H A380456 Michael De Vlieger, <a href="/A380456/b380456.txt">Table of n, a(n) for n = 1..16384</a>
%F A380456 Intersection of A131605 and A080259 = A131605 \ A055932 = A131605 \ A380446.
%e A380456 Table of n, a(n) for n = 1..12:
%e A380456  n    a(n)
%e A380456 -----------------------------
%e A380456  1    100 = 10^2 = 2^2 *  5^2
%e A380456  2    196 = 14^2 = 2^2 *  7^2
%e A380456  3    225 = 15^2 = 3^2 *  5^2
%e A380456  4    400 = 20^2 = 2^4 *  5^2
%e A380456  5    441 = 21^2 = 3^2 *  7^2
%e A380456  6    484 = 22^2 = 2^2 * 11^2
%e A380456  7    676 = 26^2 = 2^2 * 13^2
%e A380456  8    784 = 28^2 = 2^4 *  7^2
%e A380456  9   1000 = 10^3 = 2^3 *  5^3
%e A380456 10   1089 = 33^2 = 3^2 * 11^2
%e A380456 11   1156 = 34^2 = 2^2 * 17^2
%e A380456 12   1225 = 35^2 = 5^2 *  7^2
%t A380456 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[#]} &]
%Y A380456 Cf. A001597, A001694, A002110, A006530, A053669, A055932, A080259, A126706, A131605, A286708, A369417, A380446.
%K A380456 nonn
%O A380456 1,1
%A A380456 _Michael De Vlieger_, Jul 25 2025