A379733 Number of strict integer partitions of n whose product of parts is a multiple of n.

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 5, 1, 5, 7, 7, 1, 12, 1, 20, 15, 11, 1, 48, 12, 16, 33, 61, 1, 121, 1, 105, 67, 34, 126, 292, 1, 49, 128, 471, 1, 522, 1, 387, 751, 96, 1, 1556, 246, 792, 422, 869, 1, 2126, 1191, 2904, 726, 240, 1, 6393, 1, 321, 5460, 6711

Offset: 1

Author

Gus Wiseman, Jan 07 2025

Keywords

• nonn

Comments

Partitions of this type are ranked by the squarefree terms of A326149.

Examples

			The a(n) partitions for n = 1, 6, 10, 12, 15, 18:
  (1)  (6)      (10)     (12)       (15)         (18)
       (3,2,1)  (5,3,2)  (5,4,3)    (6,5,4)      (12,6)
                (5,4,1)  (6,4,2)    (7,5,3)      (9,5,4)
                         (8,3,1)    (9,5,1)      (9,6,3)
                         (6,3,2,1)  (10,3,2)     (9,7,2)
                                    (6,5,3,1)    (9,8,1)
                                    (5,4,3,2,1)  (6,5,4,3)
                                                 (7,6,3,2)
                                                 (8,6,3,1)
                                                 (9,4,3,2)
                                                 (9,6,2,1)
                                                 (12,3,2,1)
		

Crossrefs

The non-strict opposite version is A057567, ranks A326155.

The non-strict version is A057568, ranks A326149.

The case of partitions without 1's is A379735, non-strict A379734.

A319005 counts partitions with product >= sum, ranks A379721.

A114324 counts partitions with product greater than sum, ranks A325037.

Cf. A001055, A003963, A069016, A096276, A111133, A318950, A319000, A319057, A319916, A326152, A379720.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 07 2025

Mathematica

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