OFFSET
0,4
EXAMPLE
The strict partition y = (6,4,2,1) has semi-sums {3,5,6,7,8,10}, which do not include 4, so y is counted under a(13).
The a(6) = 3 through a(13) = 15 strict partitions:
(6) (7) (8) (9) (10) (11) (12) (13)
(4,2) (4,3) (5,3) (5,4) (6,4) (6,5) (7,5) (7,6)
(5,1) (5,2) (6,2) (6,3) (7,3) (7,4) (8,4) (8,5)
(6,1) (7,1) (7,2) (8,2) (8,3) (9,3) (9,4)
(4,3,1) (8,1) (9,1) (9,2) (10,2) (10,3)
(4,3,2) (5,3,2) (10,1) (11,1) (11,2)
(5,3,1) (5,4,1) (5,4,2) (5,4,3) (12,1)
(6,3,1) (6,3,2) (6,4,2) (6,4,3)
(6,4,1) (6,5,1) (6,5,2)
(7,3,1) (7,3,2) (7,4,2)
(7,4,1) (7,5,1)
(8,3,1) (8,3,2)
(5,4,2,1) (8,4,1)
(9,3,1)
(6,4,2,1)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n, 0, 15}]
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
-----------------------------------------------------------
Triangles:
A365541 counts subsets with a semi-sum k.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 19 2023
STATUS
approved