A380343 Number of strict integer partitions of n whose product of parts is a multiple of n + 1.

1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 3, 5, 5, 0, 8, 0, 15, 11, 8, 0, 42, 8, 12, 26, 49, 0, 100, 0, 90, 56, 27, 105, 246, 0, 41, 108, 414, 0, 450, 0, 332, 651, 81, 0, 1341, 210, 693, 366, 754, 0, 1869, 1044, 2579, 634, 206, 0, 5695, 0, 278, 4850, 5927, 2802

Offset: 0

Author

Gus Wiseman, Jan 22 2025

Keywords

• nonn

Examples

			The a(5) = 1 through a(17) = 8 partitions (A=10, C=12):
  32  .  421  .  54  .  83   .  76    95    843   .  98
                        632     742   653   852      863
                        641     7321  A31   861      962
                                      5432  6432     C32
                                      6521  8421     7631
                                                     9431
                                                     9521
                                                     65321
		

Crossrefs

The non-strict version is A379320, ranked by A380217 = A379319/2.

For n instead of n+1 we have A379733, non-strict A057568.

The case of equality for non-strict partitions is A380218.

A000041 counts integer partitions, strict A000009.

A379666 counts partitions by sum and product.

A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- multiple: A057567, ranks A326155

- divisor: A057568, ranks A326149

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029, A379720

- less or equal: A319005, ranks A379721, see A025147

- different: A379736, ranks A379722, see A111133

Cf. A003963, A069016, A319000, A319916, A325042, A326152, A326156, A379671, A379734.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@#,n+1]&]],{n,0,30}]