A365787 a(n) = A286708(n) divided by its squarefree kernel.

6, 12, 10, 18, 24, 14, 20, 36, 15, 48, 54, 28, 40, 72, 21, 22, 50, 96, 108, 45, 26, 56, 80, 144, 30, 44, 162, 100, 33, 75, 192, 34, 35, 216, 63, 52, 98, 38, 39, 112, 160, 288, 42, 60, 88, 324, 200, 135, 46, 384, 68, 250, 432, 51, 90, 104, 196, 76, 486, 55, 147

Offset: 1

Author

Michael De Vlieger, Sep 19 2023

Keywords

• nonn

Comments

Permutation of numbers that are not prime powers A024619.

Examples

			a(1) = 2 since b(1)/rad(b(1)) = 36/6 = 6.
a(2) = 3 since b(2)/rad(b(2)) = 72/6 = 12.
a(3) = 2 since b(3)/rad(b(3)) = 100/10 = 10.
a(4) = 4 since b(4)/rad(b(4)) = 108/6 = 18.
a(5) = 2 since b(5)/rad(b(5)) = 144/6 = 24.
a(6) = 6 since b(6)/rad(b(6)) = 196/14 = 14, etc
		

Links

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

Crossrefs

Cf. A007947, A024619, A120944, A286708, A365786.

Programs

Mathematica

nn = 5000;
s = Rest@ Select[Union@ Flatten@
  Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
  Not @* PrimePowerQ];
t = Select[Range[nn/6], And[SquareFreeQ[#], CompositeQ[#]] &];
Map[FirstPosition[t, Times @@ FactorInteger[#][[All, 1]]][[1]] &, s]

Formula

a(n) = A286708(n)/A007947(A286708(n)) = A286708(n)/A365786(n).

Let b(n) = A286708(n) and let squarefree kernel rad(n) = A007947(n). a(n) >= n such that rad(a(n)) | n.