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Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.
1

%I #9 Jan 30 2024 14:30:15

%S 108,324,648,972,1944,2700,2916,3888,4500,5832,8100,8748,9000,11664,

%T 13500,16200,17496,18000,22500,23328,24300,26244,34992,36000,40500,

%U 45000,48600,52488,67500,69984,72000,72900,78732,81000,90000,97200,104976,112500,121500,132300

%N Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.

%C Numbers k such that Omega(k) > omega(k) > 1, prime powers p^m | k are such that m > 1, rad(k) is a primorial, but k is not a product of primorials, where Omega = A001222 and omega = A001221.

%C Contains no odd numbers as a consequence of being a proper subset of A055932.

%C Proper subset of A369419, which is in turn a proper subset of A126706.

%H Michael De Vlieger, <a href="/A369420/b369420.txt">Table of n, a(n) for n = 1..10000</a>

%F {a(n)} = {A369374 \ A364930}.

%F Intersection of A056808 and A286708.

%e 36 = 2^2 * 3^2 is a product of primorials, therefore not in the sequence.

%e 72 = 2^3 * 3^2 is not a term because it is a product of primorials.

%e 100 = 2^2 * 5^2 is not in the sequence since it does not have a primorial kernel.

%e 108 = 2^2 * 3*3 is in the sequence since it is not a product of primorials, but its squarefree kernel is 6, a primorial.

%e 144 = 2^4 * 3^2 is not in the sequence since it is a product of primorials, etc.

%t With[{nn = 2^20},

%t Select[

%t Select[

%t Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],

%t Not@*PrimePowerQ],

%t And[EvenQ[#1],

%t Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, !

%t AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@

%t {#, FactorInteger[#]} &] ]

%Y Cf. A001221, A001222, A002110, A025487, A055932, A056808, A126706, A286708, A364930, A369374.

%K nonn

%O 1,1

%A _Michael De Vlieger_, Jan 22 2024