A333766 - OEIS

%I #6 Apr 06 2020 22:12:44

%S 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 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 4,4,4,4,4,3,3,3,3,3,3,3,5,4,3,3,3,2,2

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

%C One plus the longest run of 0's in the binary expansion of n.

%C 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 For n > 0, a(n) = A087117(n) + 1.

%e The 100th composition in standard order is (1,3,3), so a(100) = 3.

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[If[n==0,0,Max@@stc[n]],{n,0,100}]

%Y Positions of ones are A000225.

%Y Positions of terms <= 2 are A003754.

%Y The version for prime indices is A061395.

%Y Positions of terms > 1 are A062289.

%Y Positions of first appearances are A131577.

%Y The minimum part is given by A333768.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Compositions without 1's are A022340.

%Y - Sum is A070939.

%Y - Product is A124758.

%Y - Runs are counted by A124767.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Runs-resistance is A333628.

%Y - Weakly decreasing compositions are A114994.

%Y - Weakly increasing compositions are A225620.

%Y - Strictly decreasing compositions are A333255.

%Y - Strictly increasing compositions are A333256.

%Y Cf. A029931, A048793, A087117, A228351, A328594, A333217, A333218, A333219, A333632, A333767.

%K nonn

%O 0,3

%A _Gus Wiseman_, Apr 05 2020