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