A330972 Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.

1, 4, 8, 12, 16, 24, 36, 48, 60, 72, 96, 120, 128, 144, 180, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1440, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456, 3600

Offset: 1

Author

Gus Wiseman, Jan 06 2020

Keywords

• nonn

Comments

This is the sorted list of positions of first appearances in A001055 of each element of the range (A045782).

Examples

			Factorizations of n for n = 4, 8, 12, 16, 24, 36, 48, 60:
  4    8      12     16       24       36       48         60
  2*2  2*4    2*6    2*8      3*8      4*9      6*8        2*30
       2*2*2  3*4    4*4      4*6      6*6      2*24       3*20
              2*2*3  2*2*4    2*12     2*18     3*16       4*15
                     2*2*2*2  2*2*6    3*12     4*12       5*12
                              2*3*4    2*2*9    2*3*8      6*10
                              2*2*2*3  2*3*6    2*4*6      2*5*6
                                       3*3*4    3*4*4      3*4*5
                                       2*2*3*3  2*2*12     2*2*15
                                                2*2*2*6    2*3*10
                                                2*2*3*4    2*2*3*5
                                                2*2*2*2*3
		

Links

• R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Crossrefs

All terms belong to A025487

Includes all highly factorable numbers A033833.

Factorizations are A001055, with image A045782.

The least number with A045782(n) factorizations is A045783(n).

The least number with n factorizations is A330973(n).

The strict version is A330997.

Cf. A001222, A002033, A007716, A045778, A318284, A325238, A330935, A330936, A330976, A330977, A330989, A330991, A330992.

Programs

Mathematica

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