Multiplicative partition

In number theory, a multiplicative partition or unordered factorization of an integer ${\displaystyle n}$ is a way of writing ${\displaystyle n}$ as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number ${\displaystyle n}$ is itself considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions,^[1] which are additive partitions of finite sequences of positive integers, with the addition made pointwise. Although the study of multiplicative partitions has been ongoing since at least 1923, the name "multiplicative partition" appears to have been introduced by Hughes & Shallit (1983).^[2] The Latin name "factorisatio numerorum" had been used previously. MathWorld uses the term unordered factorization.