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A364349
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Number of strict integer partitions of n containing the sum of no subset of the parts.
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36
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1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 14, 21, 21, 28, 29, 38, 38, 51, 50, 65, 68, 82, 83, 108, 106, 130, 136, 163, 168, 206, 210, 248, 266, 307, 322, 381, 391, 457, 490, 553, 582, 675, 703, 797, 854, 952, 1000, 1147, 1187, 1331, 1437, 1564, 1656, 1869
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OFFSET
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0,4
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COMMENTS
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First differs from A275972 in counting (7,5,3,1), which is not knapsack.
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LINKS
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EXAMPLE
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The partition y = (7,5,3,1) has no subset with sum in y, so is counted under a(16).
The partition y = (15,8,4,2,1) has subset {1,2,4,8} with sum in y, so is not counted under a(31).
The a(1) = 1 through a(9) = 8 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (5,4)
(4,1) (5,1) (5,2) (6,2) (6,3)
(6,1) (7,1) (7,2)
(4,2,1) (5,2,1) (8,1)
(4,3,2)
(5,3,1)
(6,2,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Function[ptn, UnsameQ@@ptn&&Select[Subsets[ptn, {2, Length[ptn]}], MemberQ[ptn, Total[#]]&]=={}]]], {n, 0, 30}]
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CROSSREFS
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The complement in strict partitions is counted by A364272.
The linear combination-free version is A364350.
A236912 counts sum-free partitions (not re-using parts), complement A237113.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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