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A333223
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Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.
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26
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 31, 32, 33, 34, 35, 36, 40, 41, 42, 48, 50, 56, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 84, 85, 88, 96, 98, 100, 104, 106, 112, 120, 127, 128, 129, 130, 131, 132, 133
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OFFSET
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1,3
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COMMENTS
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The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.
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LINKS
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EXAMPLE
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The list of terms together with the corresponding compositions begins:
0: () 21: (2,2,1) 65: (6,1)
1: (1) 24: (1,4) 66: (5,2)
2: (2) 26: (1,2,2) 67: (5,1,1)
3: (1,1) 28: (1,1,3) 68: (4,3)
4: (3) 31: (1,1,1,1,1) 69: (4,2,1)
5: (2,1) 32: (6) 70: (4,1,2)
6: (1,2) 33: (5,1) 71: (4,1,1,1)
7: (1,1,1) 34: (4,2) 72: (3,4)
8: (4) 35: (4,1,1) 73: (3,3,1)
9: (3,1) 36: (3,3) 74: (3,2,2)
10: (2,2) 40: (2,4) 80: (2,5)
12: (1,3) 41: (2,3,1) 81: (2,4,1)
15: (1,1,1,1) 42: (2,2,2) 84: (2,2,3)
16: (5) 48: (1,5) 85: (2,2,2,1)
17: (4,1) 50: (1,3,2) 88: (2,1,4)
18: (3,2) 56: (1,1,4) 96: (1,6)
19: (3,1,1) 63: (1,1,1,1,1,1) 98: (1,4,2)
20: (2,3) 64: (7) 100: (1,3,3)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@Total/@Union[ReplaceList[stc[#], {___, s__, ___}:>{s}]]&]
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CROSSREFS
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These compositions are counted by A325676.
The number of distinct positive subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Numbers whose binary indices are a strict knapsack partition are A059519.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295 A325779, A233564, A325680, A325687, A325770, A333217.
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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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