Uniform Distribution modulo 1 of the Harmonic Prime Factor of an Integer

for the arithmetic mean of the distinct prime factors of n, then pa(n) is uniformly distributed modulo 1. Furthermore, it was also shown that for large x the number of n ≤ x such that pa(n) is an integer of order of magnitude O(x/ log log x). This was made more precise by the first author in [4], who showed that the counting function of the above n is asymptotically equal to (c+o(1))x/ log log x as x tends to infinity with some positive constant c which he did not compute. In [6], it was shown that the counting function of the composite n ≤ x fulfilling the stronger condition that pa(n) is a prime factor of n is of size x exp(−cx √ log x log log x), where 1/ √ 2 + o(1) ≤ cx ≤ 2 + o(1) as x tends to infinity. We expect similar