As requested I'm posting this an answer. I wrote a short sage script to check the primality of numbers of the form $10^n+333$ where $n$ is in the range $[4,2000]$. I found that the following values of $n$ give rise to prime numbers:

$$4,5,6,12,53,222,231,416.$$

I'm currently checking between 2000 and 3000. I'll try to write something in a bit  to check the various combinations of $10^n+3*10^i+3*10^k+3$.