cp's OEIS Frontend

A369420 Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.

%I A369420 #9 Jan 30 2024 14:30:15
%S A369420 108,324,648,972,1944,2700,2916,3888,4500,5832,8100,8748,9000,11664,
%T A369420 13500,16200,17496,18000,22500,23328,24300,26244,34992,36000,40500,
%U A369420 45000,48600,52488,67500,69984,72000,72900,78732,81000,90000,97200,104976,112500,121500,132300
%N A369420 Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.
%C A369420 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 A369420 Contains no odd numbers as a consequence of being a proper subset of A055932.
%C A369420 Proper subset of A369419, which is in turn a proper subset of A126706.
%H A369420 Michael De Vlieger, <a href="/A369420/b369420.txt">Table of n, a(n) for n = 1..10000</a>
%F A369420 {a(n)} = {A369374 \ A364930}.
%F A369420 Intersection of A056808 and A286708.
%e A369420 36 = 2^2 * 3^2 is a product of primorials, therefore not in the sequence.
%e A369420 72 = 2^3 * 3^2 is not a term because it is a product of primorials.
%e A369420 100 = 2^2 * 5^2 is not in the sequence since it does not have a primorial kernel.
%e A369420 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 A369420 144 = 2^4 * 3^2 is not in the sequence since it is a product of primorials, etc.
%t A369420 With[{nn = 2^20},
%t A369420  Select[
%t A369420    Select[
%t A369420      Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
%t A369420      Not@*PrimePowerQ],
%t A369420    And[EvenQ[#1],
%t A369420      Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, !
%t A369420      AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
%t A369420      {#, FactorInteger[#]} &] ]
%Y A369420 Cf. A001221, A001222, A002110, A025487, A055932, A056808, A126706, A286708, A364930, A369374.
%K A369420 nonn
%O A369420 1,1
%A A369420 _Michael De Vlieger_, Jan 22 2024