A369419 -id:A369419 - 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.

108, 324, 648, 972, 1944, 2700, 2916, 3888, 4500, 5832, 8100, 8748, 9000, 11664, 13500, 16200, 17496, 18000, 22500, 23328, 24300, 26244, 34992, 36000, 40500, 45000, 48600, 52488, 67500, 69984, 72000, 72900, 78732, 81000, 90000, 97200, 104976, 112500, 121500, 132300

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

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.
Contains no odd numbers as a consequence of being a proper subset of A055932.
Proper subset of A369419, which is in turn a proper subset of A126706.

			36 = 2^2 * 3^2 is a product of primorials, therefore not in the sequence.
72 = 2^3 * 3^2 is not a term because it is a product of primorials.
100 = 2^2 * 5^2 is not in the sequence since it does not have a primorial kernel.
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.
144 = 2^4 * 3^2 is not in the sequence since it is a product of primorials, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

With[{nn = 2^20},
 Select[
   Select[
     Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
     Not@*PrimePowerQ],
   And[EvenQ[#1],
     Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, !
     AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
     {#, FactorInteger[#]} &] ]

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

Intersection of A056808 and A286708.

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.

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