OFFSET
0,4
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.
EXAMPLE
For the partition y = (4,3,1) we have semi-sums {4,5,7}, which do not include 3 (the length of y), so y is counted under a(8).
The a(1) = 1 through a(8) = 16 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (311) (51) (61) (62)
(2111) (222) (322) (71)
(11111) (411) (331) (332)
(21111) (511) (422)
(111111) (4111) (431)
(22111) (611)
(31111) (4211)
(211111) (5111)
(1111111) (22211)
(221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n, 0, 10}]
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free semi-full semi-free
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Triangles:
A365541 counts subsets with a semi-sum k.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2023
STATUS
approved