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A356842
Numbers k such that the k-th composition in standard order does not cover an interval of positive integers (not gapless).
5
9, 12, 17, 19, 24, 25, 28, 33, 34, 35, 39, 40, 48, 49, 51, 56, 57, 60, 65, 66, 67, 69, 70, 71, 73, 76, 79, 80, 81, 88, 96, 97, 98, 99, 100, 103, 104, 112, 113, 115, 120, 121, 124, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 144, 145
OFFSET
1,1
COMMENTS
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.
EXAMPLE
The terms and their corresponding standard compositions begin:
9: (3,1)
12: (1,3)
17: (4,1)
19: (3,1,1)
24: (1,4)
25: (1,3,1)
28: (1,1,3)
33: (5,1)
34: (4,2)
35: (4,1,1)
39: (3,1,1,1)
40: (2,4)
48: (1,5)
49: (1,4,1)
51: (1,3,1,1)
56: (1,1,4)
57: (1,1,3,1)
60: (1,1,1,3)
MATHEMATICA
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[#]]&]
CROSSREFS
See link for sequences related to standard compositions.
An unordered version is A073492, complement A073491.
These compositions are counted by the complement of A107428.
The complement is A356841.
The gapless but non-initial version is A356843, unordered A356845.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Sequence in context: A342757 A154631 A199593 * A174525 A141552 A375241
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 01 2022
STATUS
approved