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A237668
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Number of partitions of n such that some part is a sum of two or more other parts.
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52
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0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 49, 60, 93, 115, 170, 210, 300, 370, 510, 632, 846, 1031, 1359, 1670, 2159, 2630, 3355, 4082, 5130, 6220, 7739, 9360, 11555, 13889, 16991, 20402, 24824, 29636, 35855, 42707, 51309, 60955, 72896, 86328, 102826, 121348
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OFFSET
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0,7
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COMMENTS
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These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - Gus Wiseman, Aug 12 2023
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LINKS
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EXAMPLE
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a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
The a(0) = 0 through a(9) = 13 partitions:
. . . . (211) (2111) (321) (3211) (422) (3321)
(2211) (22111) (431) (4221)
(3111) (31111) (3221) (4311)
(21111) (211111) (4211) (5211)
(22211) (32211)
(32111) (33111)
(41111) (42111)
(221111) (222111)
(311111) (321111)
(2111111) (411111)
(2211111)
(3111111)
(21111111)
(End)
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MATHEMATICA
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z = 20; m = Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #] &, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#] < 2 &)]]]]] &, IntegerPartitions[#]], False] &, Range[z]]; PartitionsP[Range[z]] - m
Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2, Length[#]}]]!={}&]], {n, 0, 15}] (* Gus Wiseman, Aug 12 2023 *)
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CROSSREFS
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These partitions have ranks A364532.
For subsets instead of partitions we have A364534, complement A151897.
A299701 counts distinct subset-sums of prime indices.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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