Uniform distribution of the fractional part of the average prime divisor

We estimate exponential sums with the function p(n) defined as the average of the prime divisors of an integer n >= 2 (we also put p(1) = 0). Our bound implies that the fractional parts of the numbers {p(n) : n >= 1} are uniformly distributed over the unit interval. We also estimate the discrepancy of the distribution, and we determine the precise order of the counting function of the set of those positive integers n such that p(n) is an integer.