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A237667
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Number of partitions of n such that no part is a sum of two or more other parts.
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57
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1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 61, 61, 87, 85, 120, 117, 160, 156, 224, 216, 288, 277, 380, 363, 483, 474, 622, 610, 783, 755, 994, 986, 1235, 1191, 1549, 1483, 1876, 1865, 2306, 2279, 2806, 2732, 3406, 3413, 4091, 4013, 4991, 4895, 5872
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OFFSET
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0,3
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COMMENTS
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Includes all knapsack partitions (A108917), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions: (4332), (5331), (33222).
These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used, ranked by A364531. The complement is counted by A237668. The binary version is A236912. For re-usable parts we have A364350.
(End)
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LINKS
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EXAMPLE
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For n = 6, the nonqualifiers are 123, 1113, 1122, 11112, leaving a(6) = 7.
The partition y = (5,3,1,1) has submultiset (3,1,1) with sum in y, so is not counted under a(10).
The partition y = (5,3,3,1) has no non-singleton submultiset with sum in y, so is counted under a(12).
The a(1) = 1 through a(8) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(311) (222) (322) (71)
(11111) (411) (331) (332)
(111111) (421) (521)
(511) (611)
(2221) (2222)
(4111) (3311)
(1111111) (5111)
(11111111)
(End)
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MATHEMATICA
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Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #]&, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#]<2&)]]]]]&, IntegerPartitions[#]], False]&, Range[20]] (* Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Total/@Subsets[#, {2, Length[#]}]]=={}&]], {n, 0, 15}] (* Gus Wiseman, Aug 09 2023 *)
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CROSSREFS
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These partitions have ranks A364531.
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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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