A370817 Greatest number of multisets that can be obtained by choosing a prime factor of each factor in an integer factorization of n into unordered factors > 1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Mar 07 2024

Keywords

• nonn

Comments

First differs from A096825 at a(210) = 4, A096825(210) = 6.

First differs from A343943 at a(210) = 4, A343943(210) = 6.

First differs from A345926 at a(90) = 4, A345926(90) = 3.

Examples

			For the factorizations of 60 we have the following choices (using prime indices {1,2,3} instead of prime factors {2,3,5}):
  (2*2*3*5): {{1,1,2,3}}
   (2*2*15): {{1,1,2},{1,1,3}}
   (2*3*10): {{1,1,2},{1,2,3}}
    (2*5*6): {{1,1,3},{1,2,3}}
    (3*4*5): {{1,2,3}}
     (2*30): {{1,1},{1,2},{1,3}}
     (3*20): {{1,2},{2,3}}
     (4*15): {{1,2},{1,3}}
     (5*12): {{1,3},{2,3}}
     (6*10): {{1,1},{1,2},{1,3},{2,3}}
       (60): {{1},{2},{3}}
So a(60) = 4.
		

Links

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

Crossrefs

For all divisors (not just prime factors) we have A370816.

The version for partitions is A370809, for all divisors A370808.

A000005 counts divisors.

A001055 counts factorizations, strict A045778.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A355741 chooses prime factors of prime indices, variations A355744, A355745.

A368413 counts non-choosable factorizations, complement A368414.

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

Cf. A000792, A048249, A066739, A319055, A319057, A355529, A355535, A367771, A368100, A370585.

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[If[#==1,{},First/@FactorInteger[#]]&/@#]]]&/@facs[n]],{n,100}]