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A356841 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).

%I A356841 #11 Sep 01 2022 19:48:36
%S A356841 0,1,2,3,4,5,6,7,8,10,11,13,14,15,16,18,20,21,22,23,26,27,29,30,31,32,
%T A356841 36,37,38,41,42,43,44,45,46,47,50,52,53,54,55,58,59,61,62,63,64,68,72,
%U A356841 74,75,77,78,82,83,84,85,86,87,89,90,91,92,93,94,95,101
%N A356841 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless).
%C A356841 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 A356841 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 A356841 The terms and their corresponding standard compositions begin:
%e A356841    0: ()
%e A356841    1: (1)
%e A356841    2: (2)
%e A356841    3: (1,1)
%e A356841    4: (3)
%e A356841    5: (2,1)
%e A356841    6: (1,2)
%e A356841    7: (1,1,1)
%e A356841    8: (4)
%e A356841   10: (2,2)
%e A356841   11: (2,1,1)
%e A356841   13: (1,2,1)
%e A356841   14: (1,1,2)
%e A356841   15: (1,1,1,1)
%e A356841   16: (5)
%e A356841   18: (3,2)
%e A356841   20: (2,3)
%e A356841   21: (2,2,1)
%t A356841 nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
%t A356841 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A356841 Select[Range[0,100],nogapQ[stc[#]]&]
%Y A356841 See link for sequences related to standard compositions.
%Y A356841 An unordered version is A073491, complement A073492.
%Y A356841 These compositions are counted by A107428.
%Y A356841 The complement is A356842.
%Y A356841 The non-initial case is A356843, unordered A356845.
%Y A356841 A356230 ranks gapless factorization lengths, firsts A356603.
%Y A356841 A356233 counts factorizations into gapless numbers.
%Y A356841 A356844 ranks compositions with at least one 1.
%Y A356841 Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224/A356225.
%K A356841 nonn
%O A356841 1,3
%A A356841 _Gus Wiseman_, Aug 31 2022