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A059519
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Number of partitions of n all of whose subpartitions sum to distinct values. Partition(n) = [a, b, c...] where 2n = 2^a + 2^b + 2^c + ...
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6
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1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 24, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 44, 48, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 80, 81, 84, 88, 96, 98, 100, 104, 112, 116, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140
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OFFSET
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1,2
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COMMENTS
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Partition encoding as in A029931. Complement of A059520.
From Gus Wiseman, Jul 22 2019: (Start)
These are numbers whose positions of 1's in their reversed binary expansion form a strict knapsack partition (A275972). The initial terms together with their corresponding partitions are:
1: (1)
2: (2)
3: (2,1)
4: (3)
5: (3,1)
6: (3,2)
8: (4)
9: (4,1)
10: (4,2)
11: (4,2,1)
12: (4,3)
14: (4,3,2)
16: (5)
17: (5,1)
18: (5,2)
19: (5,2,1)
20: (5,3)
(End)
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LINKS
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Table of n, a(n) for n=1..67.
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EXAMPLE
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14=2+4+8 so Partition(14) = [2,3,4], whose sub-sums are 0,2,3,4,5,6,7 and 14.
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MATHEMATICA
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bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[100], UnsameQ@@Total/@Subsets[bpe[#]]&] (* Gus Wiseman, Jul 22 2019 *)
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CROSSREFS
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Cf. A000120, A029931, A048793, A059520, A070939, A108917, A272020, A275972, A299702, A326015.
Other sequences classifying numbers by their binary indices: A291166 (relatively prime), A295235 (arithmetic progression), A326669 (integer average), A326675 (pairwise coprime).
Sequence in context: A298303 A333635 A102799 * A163101 A157465 A257247
Adjacent sequences: A059516 A059517 A059518 * A059520 A059521 A059522
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KEYWORD
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easy,nonn
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AUTHOR
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Marc LeBrun, Jan 19 2001
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STATUS
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approved
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