On the parity of the number of multiplicative partitions and related problems

Abstract:

Let $f(N)$ be the number of unordered factorizations of $N$, where a factorization is a way of writing $N$ as a product of integers all larger than $1$. For example, the factorizations of $30$ are \[ 2\cdot 3\cdot 5,\quad 5\cdot 6, \quad 3\cdot 10, \quad 2 \cdot 15,\quad 30, \] so that $f(30)=5$. The function $f(N)$, as a multiplicative analogue of the (additive) partition function $p(N)$, was first proposed by MacMahon, and its study was pursued by Oppenheim, Szekeres and Turán, and others.

Recently, Zaharescu and Zaki showed that $f(N)$ is even a positive proportion of the time and odd a positive proportion of the time. Here we show that for any arithmetic progression $a\operatorname {mod} m$, the set of $N$ for which \[ f(N) \equiv a( \operatorname {mod} m) \] possesses an asymptotic density. Moreover, the density is positive as long as there is at least one such $N$. For the case investigated by Zaharescu and Zaki, we show that $f$ is odd more than 50 percent of the time (in fact, about 57 percent).