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A356956
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Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).
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1
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0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376
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OFFSET
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1,3
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COMMENTS
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An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.
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LINKS
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EXAMPLE
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The terms and corresponding intervals begin:
0: ()
1: (1)
2: (2)
4: (3)
6: (1,2)
8: (4)
16: (5)
20: (2,3)
32: (6)
52: (1,2,3)
64: (7)
72: (3,4)
128: (8)
256: (9)
272: (4,5)
328: (2,3,4)
512: (10)
840: (1,2,3,4)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0, 1000], chQ[stc[#]]&]
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CROSSREFS
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See link for sequences related to standard compositions.
These compositions are counted by A001227.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
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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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