A356842 Numbers k such that the k-th composition in standard order does not cover an interval of positive integers (not gapless).

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.

Links

Table of n, a(n) for n=1..59.

Gus Wiseman, Statistics, classes, and transformations of standard 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.

Cf. A053251, A055932, A073493, A132747, A137921, A286470, A333217, A356224/A356225.

Sequence in context: A342757 A154631 A199593 * A174525 A141552 A317720

Adjacent sequences: A356839 A356840 A356841 * A356843 A356844 A356845

Keyword

nonn

Author

Gus Wiseman, Sep 01 2022

Status

approved