A380219 Number of integer partitions of n whose product is a proper multiple of n.

0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 0, 18, 0, 9, 21, 75, 0, 109, 0, 146, 83, 43, 0, 730, 224, 82, 806, 722, 0, 1782, 0, 4254, 733, 258, 1923, 9558, 0, 435, 1875, 16395, 0, 14625, 0, 9857, 33053, 1150, 0, 102070, 19391, 57326, 10157, 30702, 0, 207699, 47925, 200645

Offset: 1

Author

Gus Wiseman, Jan 21 2025

Keywords

• nonn

Examples

			The partition y = (4,3,3,2) has product 72, which is a multiple of 12, so y is counted under a(12).
The a(8) = 3 through a(14) = 9 partitions:
  (44)    (63)    (532)   .  (66)       .  (743)
  (422)   (333)   (541)      (543)         (752)
  (2222)  (3321)  (5221)     (642)         (761)
                             (831)         (7322)
                             (4332)        (7421)
                             (4431)        (72221)
                             (5322)        (73211)
                             (6222)        (74111)
                             (6321)        (722111)
                             (6411)
                             (33222)
                             (43221)
                             (43311)
                             (62211)
                             (322221)
                             (332211)
                             (432111)
                             (3222111)
		

Crossrefs

The non-proper version is A057568, case of equality A001055.

The case of strict partitions is A379733 - 1.

The case of partitions without 1's is A379734 - 1.

These partitions are ranked by A380216.

A000041 counts integer partitions, strict A000009.

A379666 counts partitions by sum and product.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- multiple: A057567, ranks A326155

- divisor: A057568 (strict A379733), ranks A326149, see A379319, A380217.

- 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, A028422, A318950, A319000, A319916, A326152, A326156, A379320, A379671, A379735, A380218, A380221.

Programs

Mathematica

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

PARI

a(n) = my(nb=0); forpart(p=n, my(vp=vecprod(Vec(p))); if (!(vp%n) && (vp>n), nb++)); nb; \\ Michel Marcus, Jan 22 2025

Formula

a(n) = A057568(n) - A001055(n).