A379679 Number of finite sets of positive integers > 1 with sum + product = n.

1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1, 0, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 0, 2, 2, 2, 2, 1, 1, 3, 0, 1, 4, 1, 1, 2, 2, 1, 2, 3, 2, 2, 0, 1, 4, 2, 1, 3, 1, 2, 2, 1, 1, 3, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 3, 2, 0, 2, 4, 3, 1, 2, 3

Offset: 1

Author

Gus Wiseman, Jan 03 2025

Keywords

• nonn

Comments

Antidiagonal sums of A379678.

Examples

			The set {2,3,4,6} has sum 15 and product 144 so is counted under a(159).
The a(n) sets for n = 47, 89, 119, 159, 179, 239:
  {5,7}    {8,9}     {2,39}  {3,39}     {2,59}   {2,79}
  {2,15}   {2,29}    {3,29}  {4,31}     {3,44}   {3,59}
  {3,11}   {4,17}    {4,23}  {7,19}     {4,35}   {4,47}
  {2,3,6}  {5,14}    {5,19}  {9,15}     {5,29}   {5,39}
           {2,3,12}  {7,14}  {2,3,22}   {8,19}   {7,29}
                     {9,11}  {2,4,17}   {9,17}   {9,23}
                             {2,7,10}   {11,14}  {11,19}
                             {2,3,4,6}           {14,15}
                                                 {2,9,12}
		

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

- strict partitions without ones: A379678, antidiagonal sums A379679 (this) (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.

A002865 counts partitions into parts > 1, strict A025147.

A316439 counts factorizations by length, partitions A008284.

A318950 counts factorizations by sum.

A326622 counts factorizations with integer mean, strict A328966.

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

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Join@@Array[facs,n],UnsameQ@@#&&Total[#]+Times@@#==n&]],{n,100}]