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A333766 Maximum part of the n-th composition in standard order. a(0) = 0.

%I A333766 #6 Apr 06 2020 22:12:44
%S A333766 0,1,2,1,3,2,2,1,4,3,2,2,3,2,2,1,5,4,3,3,3,2,2,2,4,3,2,2,3,2,2,1,6,5,
%T A333766 4,4,3,3,3,3,4,3,2,2,3,2,2,2,5,4,3,3,3,2,2,2,4,3,2,2,3,2,2,1,7,6,5,5,
%U A333766 4,4,4,4,4,3,3,3,3,3,3,3,5,4,3,3,3,2,2
%N A333766 Maximum part of the n-th composition in standard order. a(0) = 0.
%C A333766 One plus the longest run of 0's in the binary expansion of n.
%C A333766 A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
%F A333766 For n > 0, a(n) = A087117(n) + 1.
%e A333766 The 100th composition in standard order is (1,3,3), so a(100) = 3.
%t A333766 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333766 Table[If[n==0,0,Max@@stc[n]],{n,0,100}]
%Y A333766 Positions of ones are A000225.
%Y A333766 Positions of terms <= 2 are A003754.
%Y A333766 The version for prime indices is A061395.
%Y A333766 Positions of terms > 1 are A062289.
%Y A333766 Positions of first appearances are A131577.
%Y A333766 The minimum part is given by A333768.
%Y A333766 All of the following pertain to compositions in standard order (A066099):
%Y A333766 - Length is A000120.
%Y A333766 - Compositions without 1's are A022340.
%Y A333766 - Sum is A070939.
%Y A333766 - Product is A124758.
%Y A333766 - Runs are counted by A124767.
%Y A333766 - Strict compositions are A233564.
%Y A333766 - Constant compositions are A272919.
%Y A333766 - Runs-resistance is A333628.
%Y A333766 - Weakly decreasing compositions are A114994.
%Y A333766 - Weakly increasing compositions are A225620.
%Y A333766 - Strictly decreasing compositions are A333255.
%Y A333766 - Strictly increasing compositions are A333256.
%Y A333766 Cf. A029931, A048793, A087117, A228351, A328594, A333217, A333218, A333219, A333632, A333767.
%K A333766 nonn
%O A333766 0,3
%A A333766 _Gus Wiseman_, Apr 05 2020