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

0,7

Links

Table of n, a(n) for n=0..74.

Example

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.

Mathematica

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

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.

Sequence in context: A298783 A053280 A289122 * A025832 A320385 A112222

Adjacent sequences: A370806 A370807 A370808 * A370810 A370811 A370812

Keyword

nonn

Author

Gus Wiseman, Mar 05 2024

Extensions

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

Status

approved