A366741 Number of semi-sums of strict integer partitions of n.

0, 0, 0, 1, 1, 2, 5, 6, 9, 13, 21, 26, 37, 48, 63, 86, 108, 139, 175, 223, 274, 350, 422, 527, 638, 783, 939, 1146, 1371, 1648, 1957, 2341, 2770, 3285, 3867, 4552, 5353, 6262, 7314, 8529, 9924, 11511, 13354, 15423, 17825, 20529, 23628, 27116, 31139, 35615

Offset: 0

Author

Gus Wiseman, Nov 05 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 strict partitions of 9 and their a(9) = 13 semi-sums:
    (9) ->
   (81) -> 9
   (72) -> 9
   (63) -> 9
  (621) -> 3,7,8
   (54) -> 9
  (531) -> 4,6,8
  (432) -> 5,6,7
		

Crossrefs

The non-strict non-binary version is A304792.

The non-binary version is A365925.

The non-strict version is A366738.

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.

A365543 counts partitions with a subset summing to k, complement A046663.

A365661 counts strict partitions w/ subset summing to k, complement A365663.

A365924 counts incomplete partitions, ranks A365830, strict A365831.

A366739 counts semi-sums of prime indices, firsts A367097.

Cf. A008967, A122768, A299701, A364272, A364350, A365541, A365832, A366753.

Programs

Mathematica

Table[Total[Length[Union[Total/@Subsets[#, {2}]]]&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n,0,30}]