By the fundamental theorem of arithmetic, for every natural number $n$ except zero and one, there exists a function $f: \mathbb{N} \to \mathbb{N}$ such that
$$z = \prod_{i \in \mathbb{N}} p(i)^{f(i)}$$
where $p(i)$ is the $i$-th prime, counting from two as the zeroeth.
Suppose two such functions $f_1$ and $f_2$ are known. Is there an algorithm for finding the function of the sum of the numbers they correspond to, which does not involve finding their representation in a positional number system?
For application, (how) can I evaluate $$(11\cdot257\cdot1321517) + (2^2\cdot7\cdot152585957)=(5\cdot29\cdot1307\cdot42257)$$ without multiplying the summands out? Here, \begin{align*}f_1 &= \{(2, 0), (3, 0), (5, 0), \ldots, (11, 1),\ldots (1321517, 1), \ldots \}\\f_2 &= \{(2, 2), (3, 0), (5, 0), \ldots, (7,1),\ldots, (152585957, 1), \ldots \}\end{align*}