A330997 Sorted list containing the least number with each possible nonzero number of factorizations into distinct factors > 1.

1, 6, 12, 24, 48, 60, 64, 96, 120, 144, 180, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 720, 840, 864, 900, 960, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 2048, 2160, 2304, 2310, 2520, 2592, 2880, 3072, 3360, 3456, 3600, 3840, 4320

Offset: 1

Author

Gus Wiseman, Jan 06 2020

Keywords

• nonn

Examples

			The strict factorizations of a(n) for n = 1..9.
  {}  6    12   24     48     60      64     96      120
      2*3  2*6  3*8    6*8    2*30    2*32   2*48    2*60
           3*4  4*6    2*24   3*20    4*16   3*32    3*40
                2*12   3*16   4*15    2*4*8  4*24    4*30
                2*3*4  4*12   5*12           6*16    5*24
                       2*3*8  6*10           8*12    6*20
                       2*4*6  2*5*6          2*6*8   8*15
                              3*4*5          3*4*8   10*12
                              2*3*10         2*3*16  3*5*8
                                             2*4*12  4*5*6
                                                     2*3*20
                                                     2*4*15
                                                     2*5*12
                                                     2*6*10
                                                     3*4*10
                                                     2*3*4*5
		

Crossrefs

All terms belong to A025487.

Strict factorizations are A045778, with image A045779.

The unsorted version is A045780.

The non-strict version is A330972.

The least number with n strict factorizations is A330974.

Cf. A001055, A001222, A033833, A045782, A318286, A325238, A330935, A330973, A330975.

Programs

Mathematica

nn=1000;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[strfacs,nn];
Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]