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Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).
1

%I #6 Sep 26 2022 12:54:50

%S 0,1,2,4,6,8,16,20,32,52,64,72,128,256,272,328,512,840,1024,1056,2048,

%T 2320,4096,4160,8192,10512,16384,16512,17440,26896,32768,65536,65792,

%U 131072,135232,148512,262144,262656,524288,672800,1048576,1049600,1065088,1721376

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

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

%C 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.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%e The terms and corresponding intervals begin:

%e 0: ()

%e 1: (1)

%e 2: (2)

%e 4: (3)

%e 6: (1,2)

%e 8: (4)

%e 16: (5)

%e 20: (2,3)

%e 32: (6)

%e 52: (1,2,3)

%e 64: (7)

%e 72: (3,4)

%e 128: (8)

%e 256: (9)

%e 272: (4,5)

%e 328: (2,3,4)

%e 512: (10)

%e 840: (1,2,3,4)

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};

%t Select[Range[0,1000],chQ[stc[#]]&]

%Y See link for sequences related to standard compositions.

%Y These compositions are counted by A001227.

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

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

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

%Y A066311 lists gapless numbers.

%Y A356230 ranks gapless factorization lengths, firsts A356603.

%Y A356233 counts factorizations into gapless numbers.

%Y A356844 ranks compositions with at least one 1.

%Y Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.

%K nonn

%O 1,3

%A _Gus Wiseman_, Sep 24 2022