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 , , , 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.[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 is , McMahon and Oppenheim observed that its Dirichlet series generating function has the product representation[3][4]
The sequence of numbers begins
1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, ... (sequence
A001055 in the
OEIS).
Oppenheim also claimed an upper bound on , of the form[3]
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 : 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 .[6]