On asymptotic density properties of the sequence $(n!)_{n=0}^\infty $
Volume 184 / 2018
Abstract
We investigate certain arithmetic properties of factorials. On the one hand, we are interested in the densities of sets of $n$ such that the exponents of given primes in the prime factorization of $n!$ hold certain congruence properties. On the other hand, given $M$, we investigate the behavior of the $M$-free parts of factorials. In fact, we study the combination of the above two properties. We show that for any prime $p$ and positive integers $a,b$, the set of those values of $n$ for which the exponent of $p$ in $n!$ is $\alpha\pmod{p^a}$, and the $p$-free part of $n!$ is $\beta\pmod{p^b}$, has the expected density for any $\alpha,\beta$. In the particular case $p=2$, $a=1$, $b=3$, our results extend and improve a result of Deshouillers and Luca, yielding a better error term for the number of factorials up to $x$, representable as a sum of three squares.
