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A333223
Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.
26
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
OFFSET
1,3
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@Total/@Union[ReplaceList[stc[#], {___, s__, ___}:>{s}]]&]
CROSSREFS
Distinct subsequences are counted by A124770 and A124771.
A superset of A333222, counted by A169942, with partition case A325768.
These compositions are counted by A325676.
A version for partitions is A325769, with Heinz numbers A325778.
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.
Knapsack partitions are counted by A108917, with strict case A275972.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Maximal Golomb rulers are counted by A325683.
Sequence in context: A281943 A120003 A353852 * A334967 A036965 A133184
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 17 2020
STATUS
approved