A386223 -id:A386223 - cp's OEIS frontend

A380543 Nonsquarefree weak numbers k whose squarefree kernel is a primorial.

12, 18, 24, 48, 54, 60, 90, 96, 120, 150, 162, 180, 192, 240, 270, 300, 360, 384, 420, 450, 480, 486, 540, 600, 630, 720, 750, 768, 810, 840, 960, 1050, 1080, 1200, 1260, 1350, 1440, 1458, 1470, 1500, 1536, 1620, 1680, 1890, 1920, 2100, 2160, 2250, 2400, 2430

Offset: 1

Michael De Vlieger, Jul 15 2025

Numbers in this sequence have the following properties:
The number a(n) is such that rad(a(n))^2 does not divide a(n), i.e., a(n) is not powerful (i.e., in A001694), where rad = A007947.
For i > 1, prime(i) | a(n) implies prime(i-1) | a(n).

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    12   2^2 * 3
 2    18   2   * 3^2
 3    24   2^3 * 3
 4    48   2^4 * 3
 5    54   2   * 3^3
 6    60   2^2 * 3   * 5
 7    90   2   * 3^2 * 5
 8    96   2^5 * 3
 9   120   2^3 * 3   * 5
10   150   2   * 3   * 5^2
11   162   2   * 3^4
12   180   2^2 * 3^2 * 5

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

Cf. A001694, A002110, A007947, A013929, A052485, A055932, A126706, A286708, A332785, A369374, A386223, A386224.

(* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

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

The union of this sequence and A369374 is A126706.

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

Original entry on oeis.org

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

Michael De Vlieger, Jul 15 2025

			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

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

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

(* 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[#]] &]

{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.

cp's OEIS Frontend

A380543 Nonsquarefree weak numbers k whose squarefree kernel is a primorial.

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