On weighted distribution functions of sequences

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