Let $d(n)$ be the divisor function. Let $d_{2}(n)=d(d(n))$, $d_{3}(n)=d(d(d(n)))$, $d_{4}(n)=d(d(d(d(n))))$ and so on... We're gonna define $f(n)$, the smallest number satisfies $d_{f(n)}(n)=2$. For example,
$$d(12)=6$$ $$d(6)=4$$ $$d(4)=3$$ $$d(3)=2$$
hence $f(12)=3$.
Can we find an asymptotic formula for $F(n)$ where
$$F(n)=\sum_{k=1}^{n}f(k)$$