A330972 - OEIS

%I #9 Jan 07 2020 09:08:25

%S 1,4,8,12,16,24,36,48,60,72,96,120,128,144,180,192,216,240,256,288,

%T 360,384,420,432,480,576,720,768,840,864,900,960,1024,1080,1152,1260,

%U 1440,1680,1728,1800,1920,2048,2160,2304,2520,2592,2880,3072,3360,3456,3600

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

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

%H R. E. Canfield, P. Erdős and C. Pomerance, <a href="http://math.dartmouth.edu/~carlp/PDF/paper39.pdf">On a Problem of Oppenheim concerning "Factorisatio Numerorum"</a>, J. Number Theory 17 (1983), 1-28.

%e Factorizations of n for n = 4, 8, 12, 16, 24, 36, 48, 60:

%e   4    8      12     16       24       36       48         60

%e   2*2  2*4    2*6    2*8      3*8      4*9      6*8        2*30

%e        2*2*2  3*4    4*4      4*6      6*6      2*24       3*20

%e               2*2*3  2*2*4    2*12     2*18     3*16       4*15

%e                      2*2*2*2  2*2*6    3*12     4*12       5*12

%e                               2*3*4    2*2*9    2*3*8      6*10

%e                               2*2*2*3  2*3*6    2*4*6      2*5*6

%e                                        3*3*4    3*4*4      3*4*5

%e                                        2*2*3*3  2*2*12     2*2*15

%e                                                 2*2*2*6    2*3*10

%e                                                 2*2*3*4    2*2*3*5

%e                                                 2*2*2*2*3

%t nn=1000;

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t nds=Length/@Array[facs,nn];

%t Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

%Y All terms belong to A025487

%Y Includes all highly factorable numbers A033833.

%Y Factorizations are A001055, with image A045782.

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

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

%Y The strict version is A330997.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 06 2020