We derive new formulas for the number of unordered (distinct) factorizations with k parts of a positive integer n as sums over the partitions of k and an auxiliary function, the number of partitions of the prime exponents of n, where the parts have a specific number of colors. As a consequence, some new relations between partitions, Bell numbers, and Stirling numbers of the second kind are derived.

We also derive a recursive formula for the number of unordered factorizations with k different parts and a simple recursive formula for the number of partitions with k different parts.