A366738 Number of semi-sums of integer partitions of n.

0, 0, 1, 2, 5, 9, 17, 28, 46, 72, 111, 166, 243, 352, 500, 704, 973, 1341, 1819, 2459, 3277, 4363, 5735, 7529, 9779, 12685, 16301, 20929, 26638, 33878, 42778, 53942, 67583, 84600, 105270, 130853, 161835, 199896, 245788, 301890, 369208, 451046, 549002, 667370

Offset: 0

Author

Gus Wiseman, Nov 06 2023

Keywords

• nonn

Comments

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

Examples

			The partitions of 6 and their a(6) = 17 semi-sums:
       (6) ->
      (51) -> 6
      (42) -> 6
     (411) -> 2,5
      (33) -> 6
     (321) -> 3,4,5
    (3111) -> 2,4
     (222) -> 4
    (2211) -> 2,3,4
   (21111) -> 2,3
  (111111) -> 2
		

Crossrefs

The non-binary version is A304792.

The strict non-binary version is A365925.

For prime indices instead of partitions we have A366739.

The strict case is A366741.

A000041 counts integer partitions, strict A000009.

A001358 lists semiprimes, squarefree A006881, conjugate A065119.

A126796 counts complete partitions, ranks A325781, strict A188431.

A276024 counts positive subset-sums of partitions, strict A284640.

A365924 counts incomplete partitions, ranks A365830, strict A365831.

Cf. A008967, A046663, A117855, A122768, A238628, A299701, A365543, A365544, A366753, A367095.

Programs

Mathematica

Table[Total[Length[Union[Total/@Subsets[#,{2}]]]&/@IntegerPartitions[n]],{n,0,15}]

Extensions

More terms from Alois P. Heinz, Nov 06 2023