2021 Fall AMC 10A Problem 23

For each positive integer n, let f_1(n) be twice the number of positive integer divisors of n, and for j \ge 2, let f_j(n) = f_1(f_{j-1}(n)). For how many values of n \le 50 is f_{50}(n) = 12?

\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11

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