On ordered factorizations into distinct parts

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 , \displaystyle \sum _{n \le x} G(n) = x \cdot L(x)^{1+o(1)}, where \displaystyle 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).