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A356842 Numbers k such that the k-th composition in standard order does not cover an interval of positive integers (not gapless).

%I A356842 #4 Sep 01 2022 19:48:31
%S A356842 9,12,17,19,24,25,28,33,34,35,39,40,48,49,51,56,57,60,65,66,67,69,70,
%T A356842 71,73,76,79,80,81,88,96,97,98,99,100,103,104,112,113,115,120,121,124,
%U A356842 129,130,131,132,133,134,135,137,138,139,140,141,142,143,144,145
%N A356842 Numbers k such that the k-th composition in standard order does not cover an interval of positive integers (not gapless).
%C A356842 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 A356842 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 A356842 The terms and their corresponding standard compositions begin:
%e A356842    9: (3,1)
%e A356842   12: (1,3)
%e A356842   17: (4,1)
%e A356842   19: (3,1,1)
%e A356842   24: (1,4)
%e A356842   25: (1,3,1)
%e A356842   28: (1,1,3)
%e A356842   33: (5,1)
%e A356842   34: (4,2)
%e A356842   35: (4,1,1)
%e A356842   39: (3,1,1,1)
%e A356842   40: (2,4)
%e A356842   48: (1,5)
%e A356842   49: (1,4,1)
%e A356842   51: (1,3,1,1)
%e A356842   56: (1,1,4)
%e A356842   57: (1,1,3,1)
%e A356842   60: (1,1,1,3)
%t A356842 nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
%t A356842 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A356842 Select[Range[0,100],!nogapQ[stc[#]]&]
%Y A356842 See link for sequences related to standard compositions.
%Y A356842 An unordered version is A073492, complement A073491.
%Y A356842 These compositions are counted by the complement of A107428.
%Y A356842 The complement is A356841.
%Y A356842 The gapless but non-initial version is A356843, unordered A356845.
%Y A356842 A356230 ranks gapless factorization lengths, firsts A356603.
%Y A356842 A356233 counts factorizations into gapless numbers.
%Y A356842 A356844 ranks compositions with at least one 1.
%Y A356842 Cf. A053251, A055932, A073493, A132747, A137921, A286470, A333217, A356224/A356225.
%K A356842 nonn
%O A356842 1,1
%A A356842 _Gus Wiseman_, Sep 01 2022