A330973 Least positive integer with exactly n factorizations into factors > 1, and 0 if no such number exists.

1, 4, 8, 12, 16, 0, 24, 0, 36, 0, 60, 48, 0, 0, 128, 72, 0, 0, 96, 0, 120, 256, 0, 0, 0, 180, 0, 0, 144, 192, 216, 0, 0, 0, 0, 420, 0, 240, 0, 0, 0, 1024, 0, 0, 384, 0, 288, 0, 0, 0, 0, 360, 0, 0, 0, 2048, 432, 0, 0, 0, 0, 0, 0, 480, 0, 900, 768, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jan 06 2020

Keywords

• nonn

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• 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 nonzero terms belong to A025487.

Includes all highly factorable numbers A033833.

Factorizations are A001055, with image A045782.

The version without zeros is A045783.

The sorted version is A330972.

The strict version is A330974.

Positions of zeros are A330976.

Cf. A001055, A001222, A002033, A007716, A045778, A045779, A330935, A330992, A330997, A330998, A346426.

Programs

Mathematica

nn=10;
fam[n_]:=fam[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[fam[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[fam[#]&,2^nn];
Table[If[#=={},0,#[[1,1]]]&[Position[nds,i]],{i,nn}]

Extensions

More terms from Jinyuan Wang, Jul 07 2021