JBMO-SL 2020 C1

Alice and Bob play the following game: starting with the number 2 written on a blackboard, each player in turn changes the current number $n$ to a number $n+p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than $\underbrace{2 \ldots 2}_{2020}$. Assuming perfect play, who will win the game.