A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

1, 1, 2, 2, 3, 4, 5, 6, 7, 10, 11, 14, 17, 19, 23, 29, 30, 39, 41, 51, 58, 66, 78, 82, 102, 110, 132, 144, 162, 186, 210, 228, 260, 296, 328, 366, 412, 462, 512, 560, 638, 692, 764, 860, 924, 1028, 1122, 1276, 1406, 1528, 1721, 1898, 2056, 2318, 2506, 2812, 3020, 3442

Offset: 0

Author

Gus Wiseman, Mar 05 2024

Keywords

• nonn

Examples

			For the partitions of 5 we have the following choices:
      (5): {{1},{5}}
     (41): {{1,1},{1,2},{1,4}}
     (32): {{1,1},{1,2},{1,3},{2,3}}
    (311): {{1,1,1},{1,1,3}}
    (221): {{1,1,1},{1,1,2},{1,2,2}}
   (2111): {{1,1,1,1},{1,1,1,2}}
  (11111): {{1,1,1,1,1}}
So a(5) = 4.
		

Crossrefs

For just prime factors we have A370809.

The version for factorizations is A370816, for just prime factors A370817.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A006530 gives greatest prime factor, least A020639.

A027746 lists prime factors, A112798 indices, length A001222.

A239312 counts condensed partitions, ranks A368110.

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

A355733 counts choices of divisors of prime indicec.

A370320 counts non-condensed partitions, ranks A355740.

A370592 counts factor-choosable partitions, complement A370593.

Cf. A000792, A048249, A066739, A319055, A355737, A355739, A370348, A370595, A370803, A370810.

Programs

Mathematica

Table[Max[Length[Union[Sort/@Tuples[Divisors/@#]]]&/@IntegerPartitions[n]],{n,0,30}]

Extensions

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