A380543 -id:A380543 - cp's OEIS frontend

A386223 Nonsquarefree weak numbers k that are products of primorials.

12, 24, 48, 60, 96, 120, 180, 192, 240, 360, 384, 420, 480, 720, 768, 840, 960, 1080, 1260, 1440, 1536, 1680, 1920, 2160, 2520, 2880, 3072, 3360, 3840, 4320, 4620, 5040, 5760, 6144, 6300, 6480, 6720, 7560, 7680, 8640, 9240, 10080, 11520, 12288, 12600, 12960, 13440

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    12   2^2 * 3
 2    24   2^3 * 3
 3    48   2^4 * 3
 4    60   2^2 * 3 * 5
 5    96   2^5 * 3
 6   120   2^3 * 3 * 5
 7   180   2^2 * 3^2 * 5
 8   192   2^6 * 3
 9   240   2^4 * 3 * 5
10   360   2^3 * 3^2 * 5
11   384   2^7 * 3
12   420   2^2 * 3 * 5 * 7

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

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A126706, A286708, A332785, A364930, A380543.

(* Load May 19 2018 function f at A025487, then run the following: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

Subset of A380543.
Intersection of A025487 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.
The union of this sequence and A364930 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.

A386294 Nonsquarefree weak numbers k such that A053669(k) < A006530(k).

Original entry on oeis.org

20, 28, 40, 44, 45, 50, 52, 56, 63, 68, 75, 76, 80, 84, 88, 92, 98, 99, 104, 112, 116, 117, 124, 126, 132, 135, 136, 140, 147, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 184, 188, 189, 198, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 242, 244

Offset: 1

Michael De Vlieger, Jul 19 2025

nonn

			Table of n, a(n) and prime decomposition for n = 1..12:
 n  a(n)
------------------
 1   20 = 2^2 * 5
 2   28 = 2^2 * 7
 3   40 = 2^3 * 5
 4   44 = 2^2 * 11
 5   45 = 3^2 * 5
 6   50 = 2 * 5^2
 7   52 = 2^2 * 13
 8   56 = 2^3 * 7
 9   63 = 3^2 * 7
10   68 = 2^2 * 17
11   75 = 3 * 5^2
12   76 = 2^2 * 19
Let q = A053669 and let gpf = A006530.
The number 12 = 2^2*3 is not in the sequence since q(12) > gpf(12), i.e., 5 > 3.
The number 18 = 2*3^2 is not in the sequence since q(18) > gpf(18), i.e., 5 > 3.
a(1) = 20 = 2^2*5 since q(20) < gpf(20), i.e., 3 < 5.
The number 60 = 2^2*3*5 is not a term since q(60) > gpf(60), i.e., 7 > 5, etc.

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

Cf. A001694, A002110, A006939, A007947, A013929, A052485, A053669, A055932, A080259, A126706, A286708, A332785, A380543.

f[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q]; Select[Range[256], Nor[Length[#2] == 1, Max[#2[[All, -1]]] == 1, Divisible[#1, Apply[Times, #2[[All, 1]]]^2], f[#1] > #2[[-1, 1]]] & @@ {#, FactorInteger[#]} &]

Intersection of A332785 and A080259 = A332785 \ A055932 = A126706 \ A286708 \ A380543.

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A386223 Nonsquarefree weak numbers k that are products of primorials.

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

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A386294 Nonsquarefree weak numbers k such that A053669(k) < A006530(k).

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