A379734 Number of integer partitions of n into parts > 1 whose product is a multiple of n.

0, 1, 1, 2, 1, 1, 1, 4, 3, 2, 1, 8, 1, 4, 8, 27, 1, 32, 1, 40, 24, 13, 1, 175, 56, 22, 188, 166, 1, 387, 1, 874, 166, 61, 410, 1833, 1, 98, 391, 3028, 1, 2704, 1, 1828, 5893, 239, 1, 16756, 3446, 9742, 1865, 5276, 1, 32927, 8179, 31643, 3840, 814, 1, 82958, 1

Offset

1,4

Comments

Allowing 1's gives A057568.

Links

Table of n, a(n) for n=1..61.

Example

The a(n) partitions for n = 6, 10, 12, 15, 22:
  (6)  (10)     (12)         (15)         (22)
       (5,3,2)  (6,6)        (6,5,4)      (11,6,5)
                (5,4,3)      (7,5,3)      (11,7,4)
                (6,4,2)      (10,3,2)     (11,8,3)
                (4,3,3,2)    (5,4,3,3)    (11,9,2)
                (5,3,2,2)    (5,5,3,2)    (11,4,4,3)
                (6,2,2,2)    (6,5,2,2)    (11,5,4,2)
                (3,3,2,2,2)  (5,3,3,2,2)  (11,6,3,2)
                                          (11,7,2,2)
                                          (11,3,3,3,2)
                                          (11,4,3,2,2)
                                          (11,5,2,2,2)
                                          (11,3,2,2,2,2)

Maple

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=1, 1, 0), `if`(i<2, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, min(i, 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], FreeQ[#, 1]&&Divisible[Times@@#, n]&]], {n, 30}]

Crossrefs

These partitions are ranked by the odd terms of A326149.

The strict version is A379735, allowing 1's A379733.

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, A318950, A319000, A319916, A324851, A325041, A326152, A379671, A379678.

Sequence in context: A321030 A373514 A290529 * A266349 A219094 A362824

Adjacent sequences: A379731 A379732 A379733 * A379735 A379736 A379737

Keyword

nonn

Author

Gus Wiseman, Jan 07 2025

Status

approved