On the number of factorizations of an integer

Abstract

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left(C\sqrt{\frac{\log x}{\log \log x}}\left(1+o(1)\right)\right)$, where $C=2\pi \sqrt{2/3}$ and $x$ is sufficiently large. This improves upon a previous result of the first author and F. Luca.

9 pages