Structure and asymptotics for Catalan numbers modulo primes using automata

Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\{n: C_n \equiv 0 \mod p \}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \equiv -1 \mod p$. Finally we show that if $n \equiv \{ \frac{p+1}{2}, \frac{p+3}{2}, ..., p-2 \} \mod p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi.