let \[f(n) = \begin{cases} n^2+1 & \text{if }n\text{ is odd} \\ \dfrac{n}{2} & \text{if }n\text{ is even} \end{cases}. \]for how many integers $n$ from $1$ to $100$, inclusive, does $f ( f (\dotsb f (n) \dotsb )) = 1$ for some number of applications of $f$?