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A380452 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k) that are not also products of primorials, where omega = A001221.

%I A380452 #49 Aug 07 2025 18:32:48
%S A380452 324,2916,5832,8100,11664,22500,26244,72900,90000,104976,157464,
%T A380452 202500,236196,291600,360000,396900,419904,562500,656100,729000,
%U A380452 944784,1102500,1259712,1440000,1822500,1889568,2125764,2160900,2250000,2624400,3375000,3572100,3779136
%N A380452 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k) that are not also products of primorials, where omega = A001221.
%C A380452 Perfect powers k^m, m > 1, for composite k in A056808.
%C A380452 Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.
%H A380452 Michael De Vlieger, <a href="/A380452/b380452.txt">Table of n, a(n) for n = 1..16384</a>
%H A380452 Michael De Vlieger, <a href="/A380452/a380452.txt">Fast Mathematica algorithm for A055932</a>.
%F A380452 Intersection of A131605 and A056808 = A380446 \ A368682.
%F A380452 Set difference A380446 \ A025487.
%e A380452 Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading:
%e A380452                       Exponents
%e A380452  n      a(n)          2.3.5
%e A380452 -------------------------------
%e A380452  1      324 =  18^2   2.4
%e A380452  2     2916 =  54^2   2.6
%e A380452  3     5832 =  18^3   3.6
%e A380452  4     8100 =  90^2   2.4.2
%e A380452  5    11664 = 108^2   4.6
%e A380452  6    22500 = 150^2   2.2.4
%e A380452  7    26244 = 162^2   2.8
%e A380452  8    72900 = 270^2   2.6.2
%e A380452  9    90000 = 300^2   4.2.4
%e A380452 10   104976 =  18^4   4.8
%e A380452 11   157464 =  54^3   3.9
%e A380452 12   202500 = 450^2   2.4.4
%t A380452 (* Load linked Mathematica algorithm, then: *)
%t A380452 Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]
%Y A380452 Cf. A001597, A002110, A006530, A007947, A053669, A055932, A131605, A246547, A304250, A368682, A369374, A380446.
%K A380452 nonn
%O A380452 1,1
%A A380452 _Michael De Vlieger_, Jul 25 2025