A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

128, 2187, 10368, 34992, 78125, 80000, 307328, 823543, 1250000, 1366875, 1874048, 3655808, 5250987, 6328125, 10690688, 13176688, 16681088, 19487171, 32019867, 35819648, 62462907, 62748517, 66706983, 90531968, 118210688, 182660427, 187578125, 239892608, 285012027

Offset: 1

Author

Gus Wiseman, Jan 12 2020

Keywords

• nonn

Comments

Contains p^7 for all primes p.

Examples

			The 15 factorizations and 5 strict factorizations of 2187:
  (2187)           (2187)
  (27*81)          (27*81)
  (3*729)          (3*729)
  (9*243)          (9*243)
  (3*9*81)         (3*9*81)
  (9*9*27)
  (3*27*27)
  (3*3*243)
  (3*9*9*9)
  (3*3*3*81)
  (3*3*9*27)
  (3*3*3*9*9)
  (3*3*3*3*27)
  (3*3*3*3*3*9)
  (3*3*3*3*3*3*3)
		

Crossrefs

Factorizations are A001055.

Strict factorizations are A045778.

Taking "twice" instead of "three times" gives A001248.

Cf. A045779, A045780, A045782, A045783, A330972, A331023/A331024, A331048, A331050.

Programs

Mathematica

facsm[n_]:=facsm[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsm[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100000],3==Length[facsm[#]]/Length[Select[facsm[#],UnsameQ@@#&]]&]

Extensions

a(7)-(10) from Alois P. Heinz, Jan 17 2020

a(11)-a(29) from Giovanni Resta, Jan 20 2020