known combi from somewhere else (BxMO, IMOSL)

Two players play the following game:
Let $n$ be a fixed integer greater than $1$.
Starting from number $k=2$, each player has two possible moves:
either replace the number $k$ by $k+1$ or by $2k$.
The player who is forced to write a number greater than $n$ loses the game.
Which player has a winning strategy for which $n$?