A370803 Number of integer partitions of n such that more than one set can be obtained by choosing a different divisor of each part.

0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 10, 11, 15, 18, 25, 28, 39, 45, 59, 66, 83, 101, 123, 150, 176, 213, 252, 301, 352, 426, 497, 589, 684, 802, 939, 1095, 1270, 1480, 1718, 1985, 2289, 2645, 3056, 3489, 4019, 4590, 5289, 6014, 6877, 7817, 8955, 10134, 11551, 13085

Offset: 0

Author

Gus Wiseman, Mar 03 2024

Keywords

• nonn

Examples

			The partition (6,4,4,1) has two choices, namely {1,2,4,6} and {1,2,3,4}, so is counted under a(15).
The a(0) = 0 through a(13) = 18 partitions (A..D = 10..13):
  .  .  2   3   4   5    6    7    8     9     A     B     C     D
                    32   42   43   44    54    64    65    66    76
                    41        52   53    63    73    74    75    85
                              61   62    72    82    83    84    94
                                   431   81    91    92    93    A3
                                         432   433   A1    A2    B2
                                         621   532   443   543   C1
                                               541   542   633   544
                                               622   632   642   643
                                               631   641   651   652
                                                     821   732   661
                                                           741   742
                                                           822   832
                                                           831   841
                                                           921   922
                                                                 A21
                                                                 5431
                                                                 6421
		

Crossrefs

Including partitions with one choice gives A239312, complement A370320.

For a unique choice we have A370595, ranks A370810.

These partitions have ranks A370811.

A000005 counts divisors.

A000041 counts integer partitions, strict A000009.

A027746 lists prime factors, A112798 indices, length A001222.

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

A355733 counts divisor-choices of prime indices.

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

A370592 counts factor-choosable partitions, ranks A368100.

A370593 counts non-factor-choosable partitions, ranks A355529.

Cf. A355739, A368110, A370594, A370804, A370805, A370808, A370814.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]>1&]],{n,0,30}]

Formula

a(n) = A239312(n) - A370595(n). - Jinyuan Wang, Feb 14 2025

Extensions

More terms from Jinyuan Wang, Feb 14 2025