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A369636 Powerful numbers k that are neither prime powers nor products of primorials.

%I A369636 #6 Feb 03 2024 09:51:01
%S A369636 100,108,196,200,225,324,392,400,441,484,500,648,675,676,784,800,968,
%T A369636 972,1000,1089,1125,1156,1225,1323,1352,1372,1444,1521,1568,1600,1764,
%U A369636 1936,1944,2000,2025,2116,2312,2500,2601,2700,2704,2744,2888,2916,3025,3087,3136
%N A369636 Powerful numbers k that are neither prime powers nor products of primorials.
%C A369636 Proper subset of A367268 which is in turn a proper subset of A080259.
%C A369636 Superset of A369417.
%H A369636 Michael De Vlieger, <a href="/A369636/b369636.txt">Table of n, a(n) for n = 1..10000</a>
%F A369636 {a(n)} = A286708 \ A025487.
%e A369636 Let P(n) = A002110(n).
%e A369636 36 = 6^2 = P(2)^2 is a product of primorials and not in the sequence.
%e A369636 72 = 2 * 6^2 = P(1) * P(2)^2 is a product of primorials and not in the sequence.
%e A369636 a(1) = 100 = 2^2 * 5^2 is not a product of primorials.
%e A369636 a(2) = 108 = 2^2 * 3^3 is not a product of primorials, etc.
%t A369636 With[{nn = 3200}, Select[
%t A369636  Select[
%t A369636    Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
%t A369636    Not@*PrimePowerQ],
%t A369636  Nand[EvenQ[#1], Union@ Differences@ PrimePi[#2[[All, 1]]] == {1},
%t A369636    AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
%t A369636    {#, FactorInteger[#]} &] ]
%Y A369636 Cf. A001694, A002110, A025487, A080259, A126706, A286708, A367268, A369417.
%K A369636 nonn
%O A369636 1,1
%A A369636 _Michael De Vlieger_, Jan 28 2024