Let m(n) be the number of ordered factorizations of n in factors larger than 1. We prove that for every eps>0 n^{ rho} m(n) < exp[(log n)^{ 1/rho}/(loglog n)^{ 1+eps}] holds for all integers n>n_0, while, for a constant c>0, n^{ rho} m(n) > exp[c(log n)^{ 1/\rho}/(loglog n)^{ 1/rho}] holds for infinitely many positive integers n, where rho=1.72864... is the real solution to zeta(rho)=2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n).
Categories: math.NT math.CO