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 Ns 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.