Question 100
An integer is written on the blackboard. Two players $A$ and $B$ play alternately, starting with $A$. On his turn each player replaces the existing number by the result of performing one of these two operations: subtract 1 or divide by 2, as long as a positive integer is obtained. The player who reaches the number 1 wins. Determine, with justification, the smallest even number that requires $A$ to play at least 2015 times in order to win (the turns of $B$ are not counted).