2012 AMC 12B 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 $(f_1(N),f_2(N),f_3(N),\ldots )$ unbounded?
’’‘Note:’’’ A sequence of positive numbers is unbounded if for every integer $B$ , there is a member of the sequence greater than $B$ .

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.