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

0,5

Links

Table of n, a(n) for n=0..66.

Example

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)

Mathematica

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

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.

Sequence in context: A178503 A211275 A240542 * A342516 A325391 A179254

Adjacent sequences: A379662 A379664 A379666 * A379668 A379669 A379670

Keyword

nonn,new

Author

Gus Wiseman, Jan 03 2025

Status

approved