• 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 (five) 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 ${\displaystyle i}$  prime factors is the ${\displaystyle i}$ th Bell number, ${\displaystyle B_{i}}$ .

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 ${\displaystyle p^{11}}$ , ${\displaystyle p\cdot q^{5}}$ , ${\displaystyle p^{2}\cdot q^{3}}$ , and ${\displaystyle p\cdot q\cdot r^{2}}$ , where ${\displaystyle p}$ , ${\displaystyle q}$ , and ${\displaystyle r}$  are distinct prime numbers; these forms correspond to the multiplicative partitions ${\displaystyle 12}$ , ${\displaystyle 2\cdot 6}$ , ${\displaystyle 3\cdot 4}$ , and ${\displaystyle 2\cdot 2\cdot 3}$  respectively. More generally, for each multiplicative partition

${\displaystyle \prod p_{i}^{t_{i}-1},}$ 

where each ${\displaystyle p_{i}}$  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 ${\displaystyle n}$ ;^[3][4] this problem has since been studied by others under the Latin name of factorisatio numerorum. If the number of multiplicative partitions of ${\displaystyle n}$  is ${\displaystyle a_{n}}$ , McMahon and Oppenheim observed that its Dirichlet series generating function ${\displaystyle f(s)}$  has the product representation^[3][4]

The sequence of numbers ${\displaystyle a_{n}}$  begins

Oppenheim also claimed an upper bound on ${\displaystyle a_{n}}$ , of the form^[3]

Both of these bounds are not far from linear in ${\displaystyle n}$ : they are of the form ${\displaystyle n^{1-o(1)}}$ . However, the typical value of ${\displaystyle a_{n}}$  is much smaller: the average value of ${\displaystyle a_{n}}$ , averaged over an interval ${\displaystyle x\leq n\leq x+N}$ , is

Canfield, Erdős & Pomerance (1983) observe, and Luca, Mukhopadhyay & Srinivas (2010) prove, that most numbers cannot arise as the number ${\displaystyle a_{n}}$  of multiplicative partitions of some ${\displaystyle n}$ : the number of values less than ${\displaystyle N}$  which arise in this way is ${\displaystyle N^{O(\log \log \log N/\log \log N)}}$ .^[5][6] Additionally, Luca et al. show that most values of ${\displaystyle n}$  are not multiples of ${\displaystyle a_{n}}$ : the number of values ${\displaystyle n\leq N}$  such that ${\displaystyle a_{n}}$  divides ${\displaystyle n}$  is ${\displaystyle O(N/\log ^{1+o(1)}N)}$ .^[6]

• Partition (number theory)
• Divisor

• Knopfmacher, A.; Mays, M. E. (2005), "A survey of factorization counting functions" (PDF), International Journal of Number Theory, 1 (4): 563–581, doi:10.1142/S1793042105000315

• Weisstein, Eric W., "Unordered Factorization", MathWorld