摘要
Let f（n）denote the number of factorizations of the natural number n into factors larger than 1, not considering the order of the factors. We call each of these factorizations a multiplicative partition of n, or simply, partition. Also let f（1）=1. On the upper bound of f（n）, J. F. Hughes and J. O. Shallit proved that f（n）≤2n<sup>2<sup>1/2</sup></sup> and mentioned two conjectures:
