A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376

Offset: 1

Author

Gus Wiseman, Sep 24 2022

Keywords

• nonn

Comments

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

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 corresponding intervals begin:
        0: ()
        1: (1)
        2: (2)
        4: (3)
        6: (1,2)
        8: (4)
       16: (5)
       20: (2,3)
       32: (6)
       52: (1,2,3)
       64: (7)
       72: (3,4)
      128: (8)
      256: (9)
      272: (4,5)
      328: (2,3,4)
      512: (10)
      840: (1,2,3,4)
		

Links

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

Crossrefs

See link for sequences related to standard compositions.

These compositions are counted by A001227.

An unordered version is A073485, non-strict A073491 (complement A073492).

The initial version is A164894, non-strict A356843 (unordered A356845).

The non-strict version is A356841, initial A333217, counted by A107428.

A066311 lists gapless numbers.

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, A356224, A356842.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0,1000],chQ[stc[#]]&]