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

%I A380543 #19 Jul 20 2025 23:30:57
%S A380543 12,18,24,48,54,60,90,96,120,150,162,180,192,240,270,300,360,384,420,
%T A380543 450,480,486,540,600,630,720,750,768,810,840,960,1050,1080,1200,1260,
%U A380543 1350,1440,1458,1470,1500,1536,1620,1680,1890,1920,2100,2160,2250,2400,2430
%N A380543 Nonsquarefree weak numbers k whose squarefree kernel is a primorial.
%C A380543 Numbers in this sequence have the following properties:
%C A380543 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.
%C A380543 For i > 1, prime(i) | a(n) implies prime(i-1) | a(n).
%H A380543 Michael De Vlieger, <a href="/A380543/b380543.txt">Table of n, a(n) for n = 1..10000</a>
%F A380543 Intersection of A055932 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.
%F A380543 The union of this sequence and A369374 is A126706.
%e A380543 Table of n, a(n) and prime decomposition for n = 1..12:
%e A380543  n   a(n)  prime decomposition
%e A380543 ------------------------------
%e A380543  1    12   2^2 * 3
%e A380543  2    18   2   * 3^2
%e A380543  3    24   2^3 * 3
%e A380543  4    48   2^4 * 3
%e A380543  5    54   2   * 3^3
%e A380543  6    60   2^2 * 3   * 5
%e A380543  7    90   2   * 3^2 * 5
%e A380543  8    96   2^5 * 3
%e A380543  9   120   2^3 * 3   * 5
%e A380543 10   150   2   * 3   * 5^2
%e A380543 11   162   2   * 3^4
%e A380543 12   180   2^2 * 3^2 * 5
%t A380543 (* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
%t A380543 rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]
%Y A380543 Cf. A001694, A002110, A007947, A013929, A052485, A055932, A126706, A286708, A332785, A369374, A386223, A386224.
%K A380543 nonn,easy
%O A380543 1,1
%A A380543 _Michael De Vlieger_, Jul 15 2025