%I #6 Nov 18 2023 18:18:46
%S 0,0,0,0,0,1,2,5,9,13,22,30,46,63,91,118,167,216,290,374,490,626,810,
%T 1022,1297,1628,2051,2551,3176,3929,4845,5963,7311,8932,10892,13227,
%U 16035,19395,23397,28156,33803,40523,48439,57832,68876,81903,97212,115198
%N Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.
%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 a(0) = 0 through a(9) = 13 partitions:
%e . . . . . (311) (411) (331) (422) (441)
%e (3111) (421) (431) (522)
%e (511) (521) (531)
%e (4111) (611) (621)
%e (31111) (3311) (711)
%e (4211) (4311)
%e (5111) (5211)
%e (41111) (6111)
%e (311111) (33111)
%e (42111)
%e (51111)
%e (411111)
%e (3111111)
%t Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,15}]
%Y The complement for parts instead of sums is A034296, ranks A073491.
%Y The complement for all sub-sums is A126796, ranks A325781, strict A188431.
%Y For parts instead of sums we have A239955, ranks A073492.
%Y For all subset-sums we have A365924, ranks A365830, strict A365831.
%Y The complement is counted by A367402.
%Y The strict case is A367411, complement A367410.
%Y A000009 counts partitions covering an initial interval, ranks A055932.
%Y A086971 counts semi-sums of prime indices.
%Y A261036 counts complete partitions by maximum.
%Y A276024 counts positive subset-sums of partitions, strict A284640.
%Y Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365658.
%K nonn
%O 0,7
%A _Gus Wiseman_, Nov 17 2023