In number theory, a multiplicative partition or unordered factorization of an integer n 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, 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. The Latin name "factorisatio numerorum" had been used previously. MathWorld uses the term unordered factorization.

Examples

Application

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 p^{11}, p\cdot q^5,, and , where p, q, and r are distinct prime numbers; these forms correspond to the multiplicative partitions 12, 2\cdot 6, 3\cdot 4, and 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.

Bounds on the number of partitions

credits with the problem of counting the number of multiplicative partitions of n; this problem has since been studied by 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 f(s) has the product representation The sequence of numbers a_n begins Oppenheim also claimed an upper bound on a_n, of the form but as showed, this bound is erroneous and the true bound is Both of these bounds are not far from linear in n: they are of the form n^{1-o(1)}. However, the typical value of a_n is much smaller: the average value of a_n, averaged over an interval, is a bound that is of the form n^{o(1)}.

Additional results

observe, and prove, that most numbers cannot arise as the number a_n of multiplicative partitions of some n: the number of values less than N which arise in this way is. Additionally, Luca et al. show that most values of n are not multiples of a_n: the number of values n\le N such that a_n divides n is.