A379667 Number of finite multisets of positive integers with sum + product = n.

0, 1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 12, 13, 14, 15, 17, 19, 19, 20, 22, 26, 26, 29, 30, 31, 34, 35, 36, 38, 40, 43, 46, 48, 48, 50, 53, 55, 57, 61, 62, 66, 66, 69, 73, 75, 77, 79, 82, 83, 85, 89, 91, 94, 94, 95, 103, 106, 107, 111, 113, 116, 119, 121

Offset: 0

Author

Gus Wiseman, Jan 03 2025

Keywords

• nonn

Examples

			The partition (2,2,1) has sum + product equal to 5 + 4 = 9, so is counted under a(9).
The a(0) = 0 through a(8) = 5 partitions:
  .  ()  (1)  (11)  (2)    (21)    (3)      (31)      (4)
                    (111)  (1111)  (211)    (2111)    (22)
                                   (11111)  (111111)  (311)
                                                      (21111)
                                                      (1111111)
		

Crossrefs

Arrays counting multisets by sum and product:

- partitions: A379666, antidiagonal sums A379667 (this)

- partitions without ones: A379668, antidiagonal sums A379669 (zeros A379670)

- strict partitions: A379671, antidiagonal sums A379672

- strict partitions without ones: A379678, antidiagonal sums A379679 (zeros A379680)

Counting and ranking multisets by comparing sum and product:

- same: A001055 (strict A045778), ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A326156, A326172, A379733

- greater: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less: A114324, ranks A325037, see A318029

- less or equal: A319005, ranks A379721

- different: A379736, ranks A379722, see A111133

A000041 counts integer partitions, strict A000009.

A025147 counts partitions into distinct parts > 1, non-strict A002865.

A316439 counts factorizations by length, partitions A008284.

A318950 counts factorizations by sum.

A326622 counts factorizations with integer mean, strict A328966.

Cf. A003963, A028422, A069016, A319000, A319057, A319916, A325036, A325041, A326152, A326178, A379720.

Programs

Mathematica

Table[Length[Select[Join@@Array[IntegerPartitions,n+1,0],Total[#]+Times@@#==n&]],{n,0,30}]