In number theory, a multiplicative partition or unordered factorization of an integer
as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors.
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.
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
are distinct prime numbers; these forms correspond to the multiplicative partitions
More generally, for each multiplicative partition
, 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.
[2] Oppenheim (1926) credits MacMahon (1923) with the problem of counting the number of multiplicative partitions of
;[3][4] this problem has since been studied by others under the Latin name of factorisatio numerorum.
If the number of multiplicative partitions of
, McMahon and Oppenheim observed that its Dirichlet series generating function
has the product representation[3][4]
The sequence of numbers
begins Oppenheim also claimed an upper bound on
but as Canfield, Erdős & Pomerance (1983) showed, this bound is erroneous and the true bound is[5]
Both of these bounds are not far from linear in
However, the typical value of
is much smaller: the average value of
a bound that is of the form
[6] Canfield, Erdős & Pomerance (1983) observe, and Luca, Mukhopadhyay & Srinivas (2010) prove, that most numbers cannot arise as the number
of multiplicative partitions of some
[5][6] Additionally, Luca et al. show that most values of