A333768 Minimum part of the n-th composition in standard order. a(0) = 0.

0, 1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 0

Author

Gus Wiseman, Apr 06 2020

Keywords

• nonn

Comments

One plus the shortest run of 0's after a 1 in the binary expansion of n > 0.

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.

Examples

			The 148th composition in standard order is (3,2,3), so a(148) = 2.
		

Crossrefs

Positions of first appearances (ignoring index 0) are A000079.

Positions of terms > 1 are A022340.

The version for prime indices is A055396.

The maximum part is given by A333766.

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

- Length is A000120.

- Compositions without 1's are A022340.

- Sum is A070939.

- Product is A124758.

- Runs are counted by A124767.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Runs-resistance is A333628.

- Weakly decreasing compositions are A114994.

- Weakly increasing compositions are A225620.

- Strictly decreasing compositions are A333255.

- Strictly increasing compositions are A333256.

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

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Min@@stc[n]],{n,0,100}]

Formula

For n > 0, a(n) = A333767(n) + 1.