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

%I A356956 #6 Sep 26 2022 12:54:50
%S A356956 0,1,2,4,6,8,16,20,32,52,64,72,128,256,272,328,512,840,1024,1056,2048,
%T A356956 2320,4096,4160,8192,10512,16384,16512,17440,26896,32768,65536,65792,
%U A356956 131072,135232,148512,262144,262656,524288,672800,1048576,1049600,1065088,1721376
%N A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).
%C A356956 An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
%C A356956 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 A356956 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 A356956 The terms and corresponding intervals begin:
%e A356956         0: ()
%e A356956         1: (1)
%e A356956         2: (2)
%e A356956         4: (3)
%e A356956         6: (1,2)
%e A356956         8: (4)
%e A356956        16: (5)
%e A356956        20: (2,3)
%e A356956        32: (6)
%e A356956        52: (1,2,3)
%e A356956        64: (7)
%e A356956        72: (3,4)
%e A356956       128: (8)
%e A356956       256: (9)
%e A356956       272: (4,5)
%e A356956       328: (2,3,4)
%e A356956       512: (10)
%e A356956       840: (1,2,3,4)
%t A356956 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A356956 chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
%t A356956 Select[Range[0,1000],chQ[stc[#]]&]
%Y A356956 See link for sequences related to standard compositions.
%Y A356956 These compositions are counted by A001227.
%Y A356956 An unordered version is A073485, non-strict A073491 (complement A073492).
%Y A356956 The initial version is A164894, non-strict A356843 (unordered A356845).
%Y A356956 The non-strict version is A356841, initial A333217, counted by A107428.
%Y A356956 A066311 lists gapless numbers.
%Y A356956 A356230 ranks gapless factorization lengths, firsts A356603.
%Y A356956 A356233 counts factorizations into gapless numbers.
%Y A356956 A356844 ranks compositions with at least one 1.
%Y A356956 Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.
%K A356956 nonn
%O A356956 1,3
%A A356956 _Gus Wiseman_, Sep 24 2022