On ordered factorizations into distinct parts

Abstract:

Let $g(n)$ denote the number of ordered factorizations of $n$ into integers larger than $1$. In the 1930s, Kalmár and Hille investigated the average and maximal orders of $g(n)$. In this note we examine these questions for the function $G(n)$ counting ordered factorizations into distinct parts. Concerning the average of $G(n)$, we show that as $x\to \infty$, \[ \sum _{n \le x} G(n) = x \cdot L(x)^{1+o(1)}, \] where \[ L(x) = \exp \left (\log {x} \cdot \frac {\log \log \log {x}}{\log \log {x}}\right ). \] It follows immediately that $G(n) \le n \cdot L(n)^{1+o(1)}$, as $n\to \infty$. We show that equality holds here on a sequence of $n$ tending to infinity, so that $n \cdot L(n)^{1+o(1)}$ represents the maximal order of $G(n)$.