AMC 12 2012b Problem 24

Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then $$f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.$$ For every $m\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\le N\le 400$ 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 $B$.

\textbf{(A)}\ 15 \textbf{(B)}\ 16 \textbf{(C)}\ 17 \textbf{(D)}\ 18 \textbf{(E)}\ 19