A376834 Numbers k that have at least 1 powerful number m such that 1 < m <= k that are not prime powers such that rad(m) | k, where rad = A007947.

36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 100, 102, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 156, 160, 162, 168, 170, 174, 180, 186, 190, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 230, 234, 238, 240, 246, 250, 252, 255, 258, 260

Offset: 1

Author

Michael De Vlieger, Oct 06 2024

Keywords

• nonn

Comments

Numbers k such that A286708 and row k of A162306 meet.

Contains A286708, since for k in A286708, m = k is such that rad(m) | k.

Examples

			Table showing select values of a(n):
   n    a(n)                A286708 Intersect row a(n) of A162306.
  ---------------------------------------------------------------
   1    36 = 2^2 * 3^2      {36}
   2    42 = 2 * 3 * 7      {36}
   3    48 = 2^4 * 3        {36}
   4    54 = 2 * 3^3        {36}
   5    60 = 2^2 * 3 * 5    {36}
   6    66 = 2 * 3 * 11     {36}
   7    72 = 2^3 * 3^2      {36, 72}
   8    78 = 2 * 3 * 13     {36, 72}
   9    84 = 2^2 * 3 * 7    {36, 72}
  14   108 = 2^2 * 3^3      {36, 72, 108}
  17   120 = 2^3 * 3 * 5    {36, 72, 100, 108}
  24   150 = 2 * 3 * 5^2    {36, 72, 100, 108, 144}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n,pi(p)), n = 1..1500, pi = A000720, with a color function indicating exponent e = 1 in black, e = 2 = red, e = 3 = orange, ..., maximum e in magenta. The indicator bar below the image represents squarefree a(n) in green, a(n) in A332785 in blue, and a(n) in A286708 in violet.

Crossrefs

Cf. A001694, A007947, A162306, A286708.

Programs

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Reap[Do[m = 1; k = 0;
  While[Nor[k == 2, m == n + 1],
    If[And[Divisible[n, #], Divisible[m, #^2], ! PrimePowerQ[m] ] &[
      rad[m]], k++]; m++];
    If[k == 2, Sow[n]], {n, 2^10}] ][[-1, 1]]