A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 6, 4, 6, 6, 6, 6, 8, 6, 8, 8, 9, 8, 10, 9, 12, 10, 12, 12, 12, 12, 16, 13, 16, 16, 18, 16, 20, 18, 20, 20, 24, 20, 24, 24, 24, 26, 30, 26, 30, 30, 32, 32, 36, 32, 36, 36, 40, 38, 42, 40, 45, 44, 48

Offset: 0

Author

Gus Wiseman, Mar 05 2024

Keywords

• nonn

Examples

			For the partition (10,6,3,2) there are 4 choices: {2,2,2,3}, {2,2,3,3}, {2,2,3,5}, {2,3,3,5} so a(21) >= 4.
For the partitions of 6 we have the following choices:
  (6): {{2},{3}}
  (51): {}
  (42): {{2,2}}
  (411): {}
  (33): {{3,3}}
  (321): {}
  (3111): {}
  (222): {{2,2,2}}
  (2211): {}
  (21111): {}
  (111111): {}
So a(6) = 2.
		

Crossrefs

For just all divisors (not just prime factors) we have A370808.

The version for factorizations is A370817, for all divisors A370816.

A000041 counts integer partitions, strict A000009.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A355741, A355744, A355745 choose prime factors of prime indices.

A368413 counts non-choosable factorizations, complement A368414.

A370320 counts non-condensed partitions, ranks A355740.

A370592, A370593, A370594, `A370807 count non-choosable partitions.

Cf. A000792, A048249, A063834, A239312, A319055, A339095, A355529, A355733, A367771, A368100, A370585.

Programs

Mathematica

Table[Max[Length[Union[Sort /@ Tuples[If[#==1,{},First/@FactorInteger[#]]& /@ #]]]&/@IntegerPartitions[n]],{n,0,30}]

Extensions

Terms a(31) onward from Max Alekseyev, Sep 17 2024