A366738 - OEIS

%I #8 Nov 06 2023 22:58:42

%S 0,0,1,2,5,9,17,28,46,72,111,166,243,352,500,704,973,1341,1819,2459,

%T 3277,4363,5735,7529,9779,12685,16301,20929,26638,33878,42778,53942,

%U 67583,84600,105270,130853,161835,199896,245788,301890,369208,451046,549002,667370

%N Number of semi-sums of 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 partitions of 6 and their a(6) = 17 semi-sums:

%e        (6) ->

%e       (51) -> 6

%e       (42) -> 6

%e      (411) -> 2,5

%e       (33) -> 6

%e      (321) -> 3,4,5

%e     (3111) -> 2,4

%e      (222) -> 4

%e     (2211) -> 2,3,4

%e    (21111) -> 2,3

%e   (111111) -> 2

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

%Y The non-binary version is A304792.

%Y The strict non-binary version is A365925.

%Y For prime indices instead of partitions we have A366739.

%Y The strict case is A366741.

%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 A365924 counts incomplete partitions, ranks A365830, strict A365831.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 06 2023

%E More terms from _Alois P. Heinz_, Nov 06 2023