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

1,1

Comments

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.

Links

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

Formula

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

Intersection of A056808 and A286708.

Example

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.

Mathematica

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[#]} &] ]

Crossrefs

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

Sequence in context: A202609 A202100 A202492 * A202093 A202485 A202317

Adjacent sequences: A369417 A369418 A369419 * A369421 A369422 A369423

Keyword

nonn

Author

Michael De Vlieger, Jan 22 2024

Status

approved