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Sorted list containing the least number with each possible nonzero number of factorizations into distinct factors > 1.
13

%I #5 Jan 07 2020 09:08:48

%S 1,6,12,24,48,60,64,96,120,144,180,210,216,240,256,288,360,384,420,

%T 432,480,512,720,840,864,900,960,1080,1152,1260,1296,1440,1536,1680,

%U 1728,1800,2048,2160,2304,2310,2520,2592,2880,3072,3360,3456,3600,3840,4320

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

%e The strict factorizations of a(n) for n = 1..9.

%e {} 6 12 24 48 60 64 96 120

%e 2*3 2*6 3*8 6*8 2*30 2*32 2*48 2*60

%e 3*4 4*6 2*24 3*20 4*16 3*32 3*40

%e 2*12 3*16 4*15 2*4*8 4*24 4*30

%e 2*3*4 4*12 5*12 6*16 5*24

%e 2*3*8 6*10 8*12 6*20

%e 2*4*6 2*5*6 2*6*8 8*15

%e 3*4*5 3*4*8 10*12

%e 2*3*10 2*3*16 3*5*8

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

%e 2*3*20

%e 2*4*15

%e 2*5*12

%e 2*6*10

%e 3*4*10

%e 2*3*4*5

%t nn=1000;

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

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

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

%Y All terms belong to A025487.

%Y Strict factorizations are A045778, with image A045779.

%Y The unsorted version is A045780.

%Y The non-strict version is A330972.

%Y The least number with n strict factorizations is A330974.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 06 2020