On the prime power factorization of n!

In this paper we prove two results. The first theorem uses a paper of Kim \cite{K} to show that for fixed primes $p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_k$, and for fixed integers $m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_k$, with $p_i\mathrm{\mid ̸}{m}_i$, the numbers $(e_{p_1}(n),\mathrm{.}\mathrm{.}\mathrm{.},e_{p_k}(n))$ are uniformly distributed modulo $(m_1,\mathrm{.}\mathrm{.}\mathrm{.},m_k)$, where $e_p(n)$ is the order of the prime $p$ in the factorization of $n!$. That implies one of Sander’s conjecture from \cite{S}, for any set of odd primes. Berend \cite{B} asks to find the fastest growing function $f(x)$ so that for large $x$ and any given finite sequence ${ϵ}_i\in \{0,1\},i\le f(x)$, there exists $n<x$ such that the congruences $e_{p_i}(n)\equiv {ϵ}_i(\mathrm{m}\mathrm{o}\mathrm{d}2)$ hold for all $i\le f(x)$\mathrm{.}Here,$p_i$isthe$i$th prime number. In our second result, we are able to show that $f(x)$ can be taken to be at least $c_1(\log x\mathrm{/}(\log \log x{)}^6{)}^{1\mathrm{/}9}$, with some absolute constant $c_1$, provided that only the first odd prime numbers are involved.