Multiplicative partition

In number theory, a multiplicative partition or unordered factorization of an integer n that is greater than 1 is a way of writing 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 n is itself considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions, discussed in Andrews (1976), 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). The Latin name "factorisatio numerorum" had been used previously. MathWorld uses the term unordered factorization.

Hughes & Shallit (1983) describe an application of multiplicative partitions in classifying integers with a given number of divisors. For example, the integers with exactly 12 divisors take the forms p¹¹, p×q⁵, p²×q³, and p×q×r², where p, q, and r are distinct prime numbers; these forms correspond to the multiplicative partitions 12, 2×6, 3×4, and 2×2×3 respectively. More generally, for each multiplicative partition

of the integer k, there corresponds a class of integers having exactly k divisors, of the form

where each p_i is a distinct prime. This correspondence follows from the multiplicative property of the divisor function.

Oppenheim (1926) credits McMahon (1923) with the problem of counting the number of multiplicative partitions of n; this problem has since been studied by other others under the Latin name of factorisatio numerorum. If the number of multiplicative partitions of n is a_n, McMahon and Oppenheim observed that its Dirichlet series generating function ƒ(s) has the product representation

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