There is a given $N$ written on a board. Two players, Alice and Bob choose a number from the board and factorize that number to $N=XY$ where $\gcd(X,Y)\neq1$, then erase $N$ and write $X, Y$ on the board. If anyone can't factorize any number on the board, they lose. I think someone has a winning strategy in this game depending on the starting number $N$. (Zermelo's Theorem) For which $N$ does Alice have a winning strategy?
Here's what I've got. For $N=p^{2k+1}$, Bob wins.
For a $N=k^2$ for some integer $k$, Alice wins. (Factorize $N=k\cdot{k}$ then mirror Bob's factorization.)
For other $N$, who wins? Is there an deterministic algorithm to check who wins? Or is there a closed form of $N$ that Alice wins? Any help is appreciated.