cp's OEIS Frontend

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

%I A386224 #7 Jul 20 2025 23:47:15
%S A386224 18,54,90,150,162,270,300,450,486,540,600,630,750,810,1050,1200,1350,
%T A386224 1458,1470,1500,1620,1890,2100,2250,2400,2430,2940,3000,3150,3240,
%U A386224 3750,3780,4050,4200,4374,4410,4800,4860,5250,5670,5880,6000,6750,6930,7290,7350,7500
%N A386224 Nonsquarefree weak numbers k that are not products of primorials, whose squarefree kernel is a primorial.
%H A386224 Michael De Vlieger, <a href="/A386224/b386224.txt">Table of n, a(n) for n = 1..10000</a>
%F A386224 {a(n)} = A380543 \ A386223.
%F A386224 Intersection of A056808 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694, and A056808 = A055932 \ A025487.
%F A386224 The union of this sequence and A369420 is A126706.
%e A386224 Table of n, a(n) and prime decomposition for n = 1..12:
%e A386224  n   a(n)  prime decomposition
%e A386224 ------------------------------
%e A386224  1    18   2 * 3^2
%e A386224  2    54   2 * 3^3
%e A386224  3    90   2 * 3^2 * 5
%e A386224  4   150   2 * 3 * 5^2
%e A386224  5   162   2 * 3^4
%e A386224  6   270   2 * 3^3 * 5
%e A386224  7   300   2^2 * 3 * 5^2
%e A386224  8   450   2 * 3^2 * 5^2
%e A386224  9   486   2 * 3^5
%e A386224 10   540   2^2 * 3^3 * 5
%e A386224 11   600   2^3 * 3 * 5^2
%e A386224 12   630   2 * 3^2 * 5 * 7
%t A386224 (* Load May 19 2018 function f at A025487, then run the following: *)
%t A386224 fQ[x_] :=
%t A386224  Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &,
%t A386224    Nest[Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &,
%t A386224      If[x == 1, {0},
%t A386224        Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@
%t A386224          Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ x], 2] ] == x;
%t A386224 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A386224 Select[Union@ Flatten@ f[6][[3 ;; -1, 2 ;; -1]], Nor[Divisible[#, rad[#]^2], fQ[#]] &]
%Y A386224 Cf. A001694, A002110, A007947, A052485, A025487, A055932, A056808, A126706, A286708, A332785, A369420, A380543, A386223.
%K A386224 nonn,easy
%O A386224 1,1
%A A386224 _Michael De Vlieger_, Jul 15 2025