A380221 Number of strict integer partitions of n containing 1 whose product of parts is a multiple of n.

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 3, 3, 0, 4, 0, 9, 6, 4, 0, 22, 5, 6, 15, 28, 0, 54, 0, 49, 30, 14, 57, 134, 0, 22, 58, 219, 0, 242, 0, 180, 349, 44, 0, 722, 113, 369, 196, 404, 0, 994, 556, 1363, 338, 111, 0, 3016, 0, 150, 2569, 3150, 1485, 2815, 0

Offset: 1

Author

Gus Wiseman, Jan 22 2025

Keywords

• nonn

Comments

Also the number of strict integer partitions of n - 1 not containing 1 whose product of parts is a multiple of n. These are strict integer factorizations of multiples of n summing to n - 1.

Examples

			The a(6) = 1 through a(16) = 3 partitions:
  (3,2,1) . . . (5,4,1) . (8,3,1)   . (7,6,1)   (9,5,1)     (8,4,3,1)
                          (6,3,2,1)   (7,4,2,1) (6,5,3,1)   (8,5,2,1)
                                                (5,4,3,2,1) (6,4,3,2,1)
		

Crossrefs

Positions of 0 after 9 appear to be the prime numbers A000040.

The non-strict version is A379320 shifted right, ranks A380217 = A379319/2.

Not requiring 1 gives A379733.

For n instead of n+1 we have A379735 shifted left, non-strict A379734.

Partitions of this type are ranked by A379845.

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

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, A326150, A326152, A326156, A379671, A379678.

Programs

Mathematica

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