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

Author

Gus Wiseman, Sep 01 2022

Keywords

• nonn

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.

Examples

			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)
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

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.

Programs

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[#]]&]