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A356841
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Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).
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11
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0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 36, 37, 38, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 58, 59, 61, 62, 63, 64, 68, 72, 74, 75, 77, 78, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101
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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 (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 their corresponding standard compositions begin:
0: ()
1: (1)
2: (2)
3: (1,1)
4: (3)
5: (2,1)
6: (1,2)
7: (1,1,1)
8: (4)
10: (2,2)
11: (2,1,1)
13: (1,2,1)
14: (1,1,2)
15: (1,1,1,1)
16: (5)
18: (3,2)
20: (2,3)
21: (2,2,1)
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MATHEMATICA
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nogapQ[m_]:=m=={}||Union[m]==Range[Min[m], Max[m]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], nogapQ[stc[#]]&]
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CROSSREFS
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See link for sequences related to standard compositions.
These compositions are counted by A107428.
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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