A379669 Number of finite multisets of positive integers > 1 with sum + product = n.

0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 2, 2, 0, 1, 2, 4, 0, 3, 1, 1, 3, 1, 1, 2, 2, 3, 3, 2, 0, 2, 3, 2, 2, 4, 1, 4, 0, 3, 4, 2, 2, 2, 3, 1, 2, 4, 2, 3, 0, 1, 8, 3, 1, 4, 2, 3, 3, 2, 1, 3, 5, 1, 4, 3, 1, 4, 2, 7, 2, 3, 4, 3, 0, 2, 4, 6, 2, 4, 4

Offset: 0

Author

Gus Wiseman, Jan 03 2025

Keywords

• nonn

Examples

			The partition (3,2,2) has sum + product equal to 7 + 12 = 19, so is counted under a(19).
The a(n) partitions for n = 4, 8, 14, 24, 59:
  (2)  (4)    (7)      (12)       (9,5)
       (2,2)  (4,2)    (4,4)      (11,4)
              (2,2,2)  (4,2,2)    (14,3)
                       (2,2,2,2)  (19,2)
                                  (4,4,3)
                                  (11,2,2)
                                  (4,3,2,2)
                                  (3,2,2,2,2)
		

Crossrefs

Arrays counting multisets by sum and product:

- partitions: A379666, antidiagonal sums A379667

- partitions without ones: A379668, antidiagonal sums A379669 (this) (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 strict partitions into parts > 1, non-strict A002865.

A318950 counts factorizations by sum.

A326622 counts factorizations with integer mean, strict A328966.

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

Programs

Mathematica

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