Only $n \equiv 1(\bmod 210)$ work. Proof: we require $\operatorname{gcd}(n, 210)=1$. Note that $\forall p \leq 7$ the order of $n$ $(\bmod p)$ divides $p-1$, hence is relatively prime to any $p \leq 7$. So $n^{n} \equiv 1(\bmod p) \Longleftrightarrow n \equiv 1(\bmod p)$ for each of these $p$.