A340852 Numbers that can be factored in such a way that every factor is a divisor of the number of factors.

1, 4, 16, 27, 32, 64, 96, 128, 144, 192, 216, 256, 288, 324, 432, 486, 512, 576, 648, 729, 864, 972, 1024, 1296, 1458, 1728, 1944, 2048, 2560, 2592, 2916, 3125, 3888, 4096, 5120, 5184, 5832, 6144, 6400, 7776, 8192, 9216, 11664, 12288, 12800, 13824, 15552

Offset: 1

Author

Gus Wiseman, Feb 04 2021

Keywords

• nonn

Comments

Also numbers that can be factored in such a way that the length is divisible by the least common multiple.

Examples

			The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
For example, 24576 has three suitable factorizations:
  (2*2*2*2*2*2*2*2*2*2*2*12)
  (2*2*2*2*2*2*2*2*2*2*4*6)
  (2*2*2*2*2*2*2*2*2*3*4*4)
so is in the sequence.
		

Crossrefs

Partitions of this type are counted by A340693 (A340606).

These factorizations are counted by A340851.

The reciprocal version is A340853.

A143773 counts partitions whose parts are multiples of the number of parts.

A320911 can be factored into squarefree semiprimes.

A340597 have an alt-balanced factorization.

A340656 lack a twice-balanced factorization, complement A340657.

- Factorizations -

A001055 counts factorizations, with strict case A045778.

A316439 counts factorizations by product and length.

A339846 counts factorizations of even length.

A339890 counts factorizations of odd length.

A340101 counts factorizations into odd factors, odd-length case A340102.

A340653 counts balanced factorizations.

A340831/A340832 count factorizations with odd maximum/minimum.

A340785 counts factorizations into even numbers, even-length case A340786.

A340854 cannot be factored with odd least factor, complement A340855.

Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, A340830.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],And@@IntegerQ/@(Length[#]/#)&]!={}&]