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A379320
Number of integer partitions of n whose product is a multiple of n + 1.
8
1, 0, 0, 0, 0, 1, 0, 2, 2, 3, 0, 14, 0, 7, 15, 53, 0, 81, 0, 110, 61, 32, 0, 562, 170, 62, 621, 560, 0, 1400, 0, 3387, 569, 199, 1515, 7734, 0, 339, 1486, 13374, 0, 11926, 0, 8033, 27164, 913, 0, 85326, 15947, 47588, 8294, 25430, 0, 174779, 39748, 169009
OFFSET
0,8
COMMENTS
Also the number of integer partitions of n containing 1 whose product is a multiple of n. Without requiring a 1 we get A057568.
EXAMPLE
The a(5) = 1 through a(11) = 14 partitions:
(3,2) . (4,2,1) (3,3,2) (5,4) . (8,3)
(2,2,2,1) (3,3,1,1) (5,2,2) (4,4,3)
(5,2,1,1) (6,3,2)
(6,4,1)
(4,3,2,2)
(4,3,3,1)
(6,2,2,1)
(3,2,2,2,2)
(3,3,2,2,1)
(4,3,2,1,1)
(6,2,1,1,1)
(3,2,2,2,1,1)
(4,3,1,1,1,1)
(3,2,2,1,1,1,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Divisible[Times@@#, n+1]&]], {n, 0, 30}]
PROG
(PARI) a(n) = my(nb=0); forpart(p=n, if (!(vecprod(Vec(p)) % (n+1)), nb++)); nb; \\ Michel Marcus, Jan 21 2025
CROSSREFS
For n instead of n+1 we have A057568 (strict A379733), ranks A326149.
These partitions are ranked by A380217 = A379319/2 = (even case of A326149)/2.
The case of equality is A380218, see also A028422 = A001055 - 1 (ranks A325041).
A000041 counts integer partitions, strict A000009.
A379666 counts partitions by sum and product.
A380219 counts partitions of n whose product is a proper multiple of n, ranks A380216.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- multiple: A057567, ranks A326155
- divisor: A057568, ranks A326149
- 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
Sequence in context: A099118 A320999 A107098 * A293837 A181736 A322987
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 18 2025
STATUS
approved