A386224 Nonsquarefree weak numbers k that are not products of primorials, whose squarefree kernel is a primorial.

18, 54, 90, 150, 162, 270, 300, 450, 486, 540, 600, 630, 750, 810, 1050, 1200, 1350, 1458, 1470, 1500, 1620, 1890, 2100, 2250, 2400, 2430, 2940, 3000, 3150, 3240, 3750, 3780, 4050, 4200, 4374, 4410, 4800, 4860, 5250, 5670, 5880, 6000, 6750, 6930, 7290, 7350, 7500

Offset: 1

Author

Michael De Vlieger, Jul 15 2025

Keywords

• nonn
• easy

Examples

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    18   2 * 3^2
 2    54   2 * 3^3
 3    90   2 * 3^2 * 5
 4   150   2 * 3 * 5^2
 5   162   2 * 3^4
 6   270   2 * 3^3 * 5
 7   300   2^2 * 3 * 5^2
 8   450   2 * 3^2 * 5^2
 9   486   2 * 3^5
10   540   2^2 * 3^3 * 5
11   600   2^3 * 3 * 5^2
12   630   2 * 3^2 * 5 * 7
		

Links

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

Crossrefs

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A056808, A126706, A286708, A332785, A369420, A380543, A386223.

Programs

Mathematica

(* Load May 19 2018 function f at A025487, then run the following: *)
fQ[x_] :=
 Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &,
   Nest[Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &,
     If[x == 1, {0},
       Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@
         Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ x], 2] ] == x;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
Select[Union@ Flatten@ f[6][[3 ;; -1, 2 ;; -1]], Nor[Divisible[#, rad[#]^2], fQ[#]] &]

Formula

{a(n)} = A380543 \ A386223.

Intersection of A056808 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694, and A056808 = A055932 \ A025487.

The union of this sequence and A369420 is A126706.