Number of prime factors with a given multiplicity

Let k >= 1 be a natural number and omega(k) (n) denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions omega(k) with k >= 1. Moreover, we prove that the function omega(1)( n) has normal order log log n and the function (omega(1)( n) - log log n)/root log log n has a normal distribution. Finally, we prove that the functions omega(k)(n) with k >= 2 do not have normal order F (n) for any nondecreasing nonnegative function F.