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A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

%I A356843 #8 Sep 01 2022 19:48:26
%S A356843 2,4,8,10,16,18,20,32,36,42,64,68,72,74,82,84,128,136,146,148,164,170,
%T A356843 256,264,272,274,276,290,292,296,298,324,328,330,338,340,512,528,548,
%U A356843 580,584,586,594,596,658,660,676,682,1024,1040,1056,1092,1096,1098
%N A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.
%C A356843 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 A356843 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A356843 Complement of A333217 in A356841.
%e A356843 The terms together with their corresponding standard compositions begin:
%e A356843     2: (2)
%e A356843     4: (3)
%e A356843     8: (4)
%e A356843    10: (2,2)
%e A356843    16: (5)
%e A356843    18: (3,2)
%e A356843    20: (2,3)
%e A356843    32: (6)
%e A356843    36: (3,3)
%e A356843    42: (2,2,2)
%e A356843    64: (7)
%e A356843    68: (4,3)
%e A356843    72: (3,4)
%e A356843    74: (3,2,2)
%e A356843    82: (2,3,2)
%e A356843    84: (2,2,3)
%t A356843 nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
%t A356843 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A356843 Select[Range[100],!MemberQ[stc[#],1]&&nogapQ[stc[#]]&]
%Y A356843 See link for sequences related to standard compositions.
%Y A356843 A subset of A022340.
%Y A356843 These compositions are counted by A251729.
%Y A356843 The unordered version (using Heinz numbers of partitions) is A356845.
%Y A356843 A333217 ranks complete compositions.
%Y A356843 A356230 ranks gapless factorization lengths, firsts A356603.
%Y A356843 A356233 counts factorizations into gapless numbers.
%Y A356843 A356841 ranks gapless compositions, counted by A107428.
%Y A356843 A356842 ranks non-gapless compositions, counted by A356846.
%Y A356843 A356844 ranks compositions with at least one 1.
%Y A356843 Cf. A053251, A055932, A073491, A073492, A073493, A137921, A356224/A356225.
%K A356843 nonn
%O A356843 1,1
%A A356843 _Gus Wiseman_, Sep 01 2022