Consider a function $g:\mathbb N\to\mathbb N$ such that $g(g(n))=2n$. (I think it is well-known that it exists. If not, then see here.) Now, if the prime factorization of $n$ is $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$, then, one such function $F$ is: $$F(n)=p_1^{g(e_1)}p_2^{g(e_2)}\cdots p_k^{g(e_k)}$$(Note that $F(1)=1$.)