  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 number 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.
