2020 AMC 12B Problem 24

Let D(n) denote the number of ways of writing the positive integer n as a productn = f_1\cdot f_2\cdots f_k,where k\ge1, the f_i are integers strictly greater than 1, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number 6 can be written as 6, 2\cdot 3, and 3\cdot2, so D(6) = 3. What is D(96)?

\textbf{(A) } 112 \qquad\textbf{(B) } 128 \qquad\textbf{(C) } 144 \qquad\textbf{(D) } 172 \qquad\textbf{(E) } 184

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