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

1,12

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.

Links

Table of n, a(n) for n=1..67.

Example

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)

Mathematica

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

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.

Sequence in context: A002125 A375201 A171731 * A323212 A185815 A332448

Adjacent sequences: A380218 A380219 A380220 * A380222 A380223 A380224

Keyword

nonn

Author

Gus Wiseman, Jan 22 2025

Status

approved