ON THE PARITY OF THE NUMBER OF MULTIPLICATIVE PARTITIONS AND RELATED PROBLEMS

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 2.3.5, 5.6, 3.10, 2.15, 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 Turin, 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 mod m, the set of N for which f(N) equivalent to a(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).