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 A059519 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 + ... 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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) LINKS EXAMPLE 14=2+4+8 so Partition(14) = [2,3,4], whose sub-sums are 0,2,3,4,5,6,7 and 14. MATHEMATICA bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1]; Select[Range[100], UnsameQ@@Total/@Subsets[bpe[#]]&] (* Gus Wiseman, Jul 22 2019 *) CROSSREFS 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 KEYWORD easy,nonn AUTHOR Marc LeBrun, Jan 19 2001 STATUS approved

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Last modified March 24 08:04 EDT 2023. Contains 361455 sequences. (Running on oeis4.)