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A364613
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a(n) = number of partitions of n whose sum multiset is free of duplicates; see Comments.
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0
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1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 18, 20, 26, 29, 36, 38, 50, 53, 67, 69, 89, 95, 115, 122, 151, 161, 195, 201, 247, 266, 312, 330, 386, 419, 487, 520, 600, 641, 742, 793, 901, 979, 1088, 1186, 1331, 1454, 1605, 1730, 1925, 2102, 2311, 2525, 2741, 3001
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OFFSET
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0,3
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COMMENTS
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If M is a multiset of real numbers, then the sum multiset of M consists of the sums of pairs of distinct numbers in M. For example, the sum multiset of (1,2,4,5) is {3,5,6,6,7,9}.
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LINKS
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FORMULA
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EXAMPLE
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The partitions of 8 are [8], [7,1], [6,2], [6,1,1], [5,3], [5,2,1], [5,1,1,1], [4,4], [4,3,1], [4,2,2], [4,2,1,1], [4,1,1,1,1], [3,3,2], [3,3,1,1], [3,2,2,1], [3,2,1,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1]. The 7 partitions whose sum multiset is duplicate-free are [8], [7,1], [6,2], [5,3], [5,2,1], [4,4], [4,3,1].
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MATHEMATICA
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s[n_, k_] := s[n, k] = Subsets[IntegerPartitions[n][[k]], {2}];
g[n_, k_] := g[n, k] = DuplicateFreeQ[Map[Total, s[n, k]]];
t[n_] := Table[g[n, k], {k, 1, PartitionsP[n]}];
a[n_] := Count[t[n], True]
Table[a[n], {n, 1, 40}]
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CROSSREFS
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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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