On the number of multiplicative partitions. (English) Zbl 0523.10007

Hughes, John F.; Shallit, J. O.

Am. Math. Mon. 90, 468-471 (1983).

In this paper the authors have defined multiplicative partitions of \(n\) as follows: Let \(f(n)\) denote the number of ways to represent \(n\) as the product of integers. Then \(f(n)\) is called the multiplicative partition function of \(n\). If \(q\) is prime then \(f(q^k ) = p(k)\), the number of additive partitions of \(k\) and if \(q_1,q_2 \cdots q_k\) are distinct primes then \(f(q_1,\ldots,q_k)=B(k)\), the \(k\)-th Bell number. The authors prove the estimate: \(f(n) \le 2n^{\sqrt{2}}\) with the help of an auxiliary function \(g(m,n)\) defined by them as the number of multiplicative partitions of \(n\) with all elements \(\le m\).
They further make two conjectures: \(f(n)\le n\) and \(f(n)\le n/\log n\) for \(n\ne144\) which are verified by them on a computer for \(n\le 10,000\).

Reviewer: Indulata Sukla (Orissa)

MSC:

11P81

Elementary theory of partitions

Keywords:

multiplicative partition function; additive partition; Bell number