A370816 Greatest number of multisets that can be obtained by choosing a divisor of each factor in an integer factorization of n into unordered factors > 1.

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 2, 4, 4, 7, 2, 7, 2, 7, 4, 4, 2, 11, 3, 4, 5, 7, 2, 8, 2, 10, 4, 4, 4, 12, 2, 4, 4, 11, 2, 8, 2, 7, 7, 4, 2, 17, 3, 7, 4, 7, 2, 11, 4, 11, 4, 4, 2, 15, 2, 4, 7, 14, 4, 8, 2, 7, 4, 8, 2, 20, 2, 4, 7, 7, 4, 8, 2, 17, 7, 4, 2

Offset: 1

Author

Gus Wiseman, Mar 06 2024

Keywords

• nonn

Examples

			For the factorizations of 12 we have the following choices:
  (2*2*3): {{1,1,1},{1,1,2},{1,1,3},{1,2,2},{1,2,3},{2,2,3}}
    (2*6): {{1,1},{1,2},{1,3},{1,6},{2,2},{2,3},{2,6}}
    (3*4): {{1,1},{1,2},{1,3},{1,4},{2,3},{3,4}}
     (12): {{1},{2},{3},{4},{6},{12}}
So a(12) = 7.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..10000

Crossrefs

The version for partitions is A370808, for just prime factors A370809.

For just prime factors we have A370817.

A000005 counts divisors.

A001055 counts factorizations, strict A045778.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A368413 counts non-choosable factorizations, complement A368414.

A370813 counts non-divisor-choosable factorizations, complement A370814.

Cf. A000792, A048249, A066739, A319055, A319057, A355737, A355739, A355740, A355741, A368110, A370803.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@facs[n]],{n,100}]