ISL 2004 C5

$A$ and $B$ play a game, given an integer $N$ , $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$ , but his number cannot be bigger than $N$ . The player who writes $N$ wins. For which values of $N$ does $B$ win?

Proposed by A. Slinko & S. Marshall, New Zealand