OPEN
Are there infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $105$?
Erdős, Graham, Ruzsa, and Straus
[EGRS75] have shown that, for any two odd primes $p$ and $q$, there are infinitely many $n$ such that $\binom{2n}{n}$ is coprime to $pq$.
This is equivalent (via Kummer's theorem) to whether there are infinitely many $n$ which have only digits $0,1$ in base $3$, digits $0,1,2$ in base $5$, and digits $0,1,2,3$ in base $7$.
The sequence of such $n$ is A030979 in the OEIS.