We're given factorization of an integer $x=p_1 \cdot p_2 \cdots p_k$ where $p_1<p_2<\cdots<p_k$ are prime numbers such that the power of any prime numbers is not greater than $1$. For example $30=2\cdot 3\cdot 5$ is legal input while $300 = 2^2\cdot 3\cdot 5$ is not a legal input.
Let $f(x)$ be the number of ways to represent $x$ as a product of integers (not necessarily prime numbers). For example, $f(30)=5$ because $$30=30\cdot 1=2\cdot 3\cdot 5=2\cdot 15=3\cdot 10=5\cdot 6$$
(order is not important).
What could be an algorithm which calculates $f(x)$ given $x=p_1 \cdot p_2 \cdots p_k$?
Ideally the algorithm should use dynamic programming principles but most importantly what would be the recursive formula to find $f(x)$?