A379735 Number of strict integer partitions of n into parts > 1 whose product is a multiple of n.

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 3, 4, 4, 1, 8, 1, 11, 9, 7, 1, 26, 7, 10, 18, 33, 1, 67, 1, 56, 37, 20, 69, 158, 1, 27, 70, 252, 1, 280, 1, 207, 402, 52, 1, 834, 133, 423, 226, 465, 1, 1132, 635, 1541, 388, 129, 1, 3377, 1, 171, 2891, 3561, 1674, 3154

Offset

1,9

Comments

These partitions are ranked by the odd squarefree terms of A326149.

Links

Table of n, a(n) for n=1..66.

Example

The a(n) partitions for n = 2, 9, 12, 15, 18, 20, 21:
  (2)  (9)    (12)     (15)      (18)       (20)         (21)
       (6,3)  (5,4,3)  (6,5,4)   (12,6)     (8,7,5)      (8,7,6)
              (6,4,2)  (7,5,3)   (9,5,4)    (10,6,4)     (9,7,5)
                       (10,3,2)  (9,6,3)    (10,8,2)     (11,7,3)
                                 (9,7,2)    (11,5,4)     (12,7,2)
                                 (6,5,4,3)  (12,5,3)     (14,4,3)
                                 (7,6,3,2)  (7,6,5,2)    (7,6,5,3)
                                 (9,4,3,2)  (8,5,4,3)    (9,7,3,2)
                                            (9,5,4,2)    (7,5,4,3,2)
                                            (10,5,3,2)
                                            (6,5,4,3,2)

Mathematica

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

Crossrefs

Allowing 1's gives A379733.

The non-strict version is A379734, allowing 1's A057568.

A000041 counts integer partitions, strict A000009.

A002865 counts partitions into parts > 1.

A379666 counts partitions by sum and product, without 1's A379668.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A379733

- 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. A069016, A319000, A319057, A319916, A324851, A326152, A379671, A379678.

Sequence in context: A387865 A044924 A374112 * A057036 A394355 A069004

Adjacent sequences: A379732 A379733 A379734 * A379736 A379737 A379738

Keyword

nonn

Author

Gus Wiseman, Jan 07 2025

Status

approved