OME 2015 Problem 3

On the board is written an integer $N \geq 2$ . Two players $A$ and $B$ play in turn, starting with $A$ . Each player in turn replaces the existing number by the result of performing one of two operations: subtract 1 and divide by 2, provided that a positive integer is obtained. The player who reaches the number 1 wins. Determine the smallest even number $N$ requires you to play at least $2015$ times to win ( $B$ shifts are not counted).