%I #4 Sep 01 2022 19:48:31

%S 9,12,17,19,24,25,28,33,34,35,39,40,48,49,51,56,57,60,65,66,67,69,70,

%T 71,73,76,79,80,81,88,96,97,98,99,100,103,104,112,113,115,120,121,124,

%U 129,130,131,132,133,134,135,137,138,139,140,141,142,143,144,145

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

%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>

%e The terms and their corresponding standard compositions begin:

%e 9: (3,1)

%e 12: (1,3)

%e 17: (4,1)

%e 19: (3,1,1)

%e 24: (1,4)

%e 25: (1,3,1)

%e 28: (1,1,3)

%e 33: (5,1)

%e 34: (4,2)

%e 35: (4,1,1)

%e 39: (3,1,1,1)

%e 40: (2,4)

%e 48: (1,5)

%e 49: (1,4,1)

%e 51: (1,3,1,1)

%e 56: (1,1,4)

%e 57: (1,1,3,1)

%e 60: (1,1,1,3)

%t nogapQ[m_]:=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[0,100],!nogapQ[stc[#]]&]

%Y See link for sequences related to standard compositions.

%Y An unordered version is A073492, complement A073491.

%Y These compositions are counted by the complement of A107428.

%Y The complement is A356841.

%Y The gapless but non-initial version is A356843, unordered A356845.

%Y A356230 ranks gapless factorization lengths, firsts A356603.

%Y A356233 counts factorizations into gapless numbers.

%Y A356844 ranks compositions with at least one 1.

%Y Cf. A053251, A055932, A073493, A132747, A137921, A286470, A333217, A356224/A356225.

%K nonn

%O 1,1

%A _Gus Wiseman_, Sep 01 2022