Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Nov 07 2023 08:24:54
%S 0,0,0,1,1,2,5,6,9,13,21,26,37,48,63,86,108,139,175,223,274,350,422,
%T 527,638,783,939,1146,1371,1648,1957,2341,2770,3285,3867,4552,5353,
%U 6262,7314,8529,9924,11511,13354,15423,17825,20529,23628,27116,31139,35615
%N Number of semi-sums of strict integer partitions of n.
%C 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.
%e The strict partitions of 9 and their a(9) = 13 semi-sums:
%e (9) ->
%e (81) -> 9
%e (72) -> 9
%e (63) -> 9
%e (621) -> 3,7,8
%e (54) -> 9
%e (531) -> 4,6,8
%e (432) -> 5,6,7
%t Table[Total[Length[Union[Total/@Subsets[#, {2}]]]&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n,0,30}]
%Y The non-strict non-binary version is A304792.
%Y The non-binary version is A365925.
%Y The non-strict version is A366738.
%Y A000041 counts integer partitions, strict A000009.
%Y A001358 lists semiprimes, squarefree A006881, conjugate A065119.
%Y A126796 counts complete partitions, ranks A325781, strict A188431.
%Y A276024 counts positive subset-sums of partitions, strict A284640.
%Y A365543 counts partitions with a subset summing to k, complement A046663.
%Y A365661 counts strict partitions w/ subset summing to k, complement A365663.
%Y A365924 counts incomplete partitions, ranks A365830, strict A365831.
%Y A366739 counts semi-sums of prime indices, firsts A367097.
%Y Cf. A008967, A122768, A299701, A364272, A364350, A365541, A365832, A366753.
%K nonn
%O 0,6
%A _Gus Wiseman_, Nov 05 2023