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A356843
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Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.
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6
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2, 4, 8, 10, 16, 18, 20, 32, 36, 42, 64, 68, 72, 74, 82, 84, 128, 136, 146, 148, 164, 170, 256, 264, 272, 274, 276, 290, 292, 296, 298, 324, 328, 330, 338, 340, 512, 528, 548, 580, 584, 586, 594, 596, 658, 660, 676, 682, 1024, 1040, 1056, 1092, 1096, 1098
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OFFSET
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1,1
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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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FORMULA
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EXAMPLE
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The terms together with their corresponding standard compositions begin:
2: (2)
4: (3)
8: (4)
10: (2,2)
16: (5)
18: (3,2)
20: (2,3)
32: (6)
36: (3,3)
42: (2,2,2)
64: (7)
68: (4,3)
72: (3,4)
74: (3,2,2)
82: (2,3,2)
84: (2,2,3)
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MATHEMATICA
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nogapQ[m_]:=Or[m=={}, Union[m]==Range[Min[m], Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[100], !MemberQ[stc[#], 1]&&nogapQ[stc[#]]&]
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CROSSREFS
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See link for sequences related to standard compositions.
These compositions are counted by A251729.
The unordered version (using Heinz numbers of partitions) is A356845.
A333217 ranks complete compositions.
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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