A379672 Number of finite sets of positive integers with sum + product = n.

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

Offset: 0

Author

Gus Wiseman, Jan 03 2025

Keywords

• nonn

Comments

Antidiagonal sums of A379671, starting with 0.

The only zeros are a(0) and a(3).

Examples

			The a(n) sets for n = 2, 11, 20, 35, 47, 60:
  {1}  {1,5}  {10}     {3,8}    {5,7}    {30}
       {2,3}  {2,6}    {1,17}   {1,23}   {1,5,9}
              {1,3,4}  {2,11}   {2,15}   {2,4,6}
                       {1,4,6}  {3,11}   {1,2,19}
                                {2,3,6}  {1,3,14}
                                         {1,4,11}
		

Crossrefs

Arrays counting multisets by sum and product:

- partitions: A379666, antidiagonal sums A379667

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

- strict partitions: A379671, antidiagonal sums A379672 (this)

- 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.

Cf. A003963, A028422, A096765, A319000, A319057, A319916, A325036, A326152, A326178.

Programs

Mathematica

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

Extensions

More terms from Jinyuan Wang, Jan 11 2025