Define the function $f_{1}$ on the positive integers by setting $f_{1} (1) = 1$ and if $n = p_{1}^{e_{1}} p_{2}^{e_{2}} \dots p_{k}^{e_{k}}$ is the prime factorization of $n > 1$ , then

$f_{1} (n) = (p_{1} + 1)^{e_{1} - 1} (p_{2} + 1)^{e_{2} - 1} \dots (p_{k} + 1)^{e_{k} - 1}$ .

For every $m \geq 2$ , let $f_{m} (n) = f_{1} (f_{m - 1} (n))$ . For how many $N$ s in the range $1 \leq N \leq 500$ is the sequence $(f_{1} (N), f_{2} (N), f_{3} (N), \dots$ ) unbounded?

Note: A sequence of positive numbers is unbounded if for every integer $B$ , there is a member of the sequence greater than $B$ .