OFFSET
0,4
COMMENTS
The Heinz numbers of these partitions are given by A367226.
EXAMPLE
The partition (4,2,1) has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(7).
The a(1) = 1 through a(7) = 13 partitions:
(1) (11) (21) (22) (32) (42) (52)
(111) (31) (41) (51) (61)
(211) (221) (321) (322)
(1111) (311) (411) (331)
(2111) (2211) (421)
(11111) (3111) (511)
(21111) (2221)
(111111) (3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], combs[Length[#], Union[#]]!={}&]], {n, 0, 15}]
CROSSREFS
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
sum-full sum-free comb-full comb-free
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KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 14 2023
EXTENSIONS
a(31)-a(48) from Chai Wah Wu, Nov 15 2023
STATUS
approved