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

1,6

Comments

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

Links

Table of n, a(n) for n=1..64.

Example

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)

Maple

b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
      `if`(n=0, `if`(t=1, 1, 0), b(n, i-1, t)+
         b(n-i, min(i-1, n-i), t/igcd(i, t))))
    end:
a:= n-> `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}]

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.

Sequence in context: A379003 A217612 A029254 * A063740 A072782 A337014

Adjacent sequences: A379730 A379731 A379732 * A379734 A379735 A379736

Keyword

nonn

Author

Gus Wiseman, Jan 07 2025

Status

approved