Multiplicative partition


In number theory, a multiplicative partition or unordered factorization of an integer is a way of writing 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 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

  • The number 20 has four multiplicative partitions: 2 × 2 × 5, 2 × 10, 4 × 5, and 20.
  • 3 × 3 × 3 × 3, 3 × 3 × 9, 3 × 27, 9 × 9, and 81 are the five multiplicative partitions of 81 = 34. Because it is the fourth power of a prime, 81 has the same number of multiplicative partitions as 4 does of additive partitions.
  • The number 30 has five multiplicative partitions: 2 × 3 × 5 = 2 × 15 = 6 × 5 = 3 × 10 = 30.
  • In general, the number of multiplicative partitions of a squarefree number with prime factors is the th Bell number,.

    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,,, and, where,, and are distinct prime numbers; these forms correspond to the multiplicative partitions,,, and respectively. More generally, for each multiplicative partition
of the integer, there corresponds a class of integers having exactly divisors, of the form
where each 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 ; this problem has since been studied by others under the Latin name of factorisatio numerorum. If the number of multiplicative partitions of is, McMahon and Oppenheim observed that its Dirichlet series generating function has the product representation
The sequence of numbers begins
Oppenheim also claimed an upper bound on, 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 : they are of the form.
However, the typical value of is much smaller: the average value of, averaged over an interval, is
a bound that is of the form.

Additional results

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