On weighted distribution functions of sequences

In this paper we prove that the set of logarithmically weighted distribution functions of the sequence of iterated logarithm $log^{(i)} n mod 1$, $n = n_i, n_i + 1\dots$ is the same as the set of classical distribution functions of the sequence $log^{(i+1)} n$ mod 1 for every $i = 2, 3, \dots$ . Also we prove that $log(n log n)$ mod 1 is logarithmically uniformly distributed. This implies that the sequence $p_n/n$ mod 1, where $p_n $denotes the nth prime, is also logarithmically uniformly distributed.