The distribution of numbers with many ordered factorizations

Let g(n) be the number of ordered factorizations of n into numbers larger than 1. We find precise bounds on the positive moments of g. We use these results to estimate the number of n <= x satisfying g(n) >= x(alpha) for all positive alpha. In addition, let G(n) and gP(n) be the number of ordered factorizations of n into distinct numbers larger than 1 and primes, respectively. We also bound the positive moments of G and gP from below.