$101$ is the only prime in the sequence $1,101,10101,\ldots$ as shown in this Putnam question.
I also know from studying the Collatz conjecture that $101_2$ is also the only prime in the same sequence considered as a sequence of base $2$ numbers.
In what bases is this true, and is there a general proof for some class of bases?
The proof in the Putnam question goes through regardless of the base. Replace $10$ by $b$, $9$ by $b-1$ and $11$ by $b+1$. It is true in all bases $b$ where $b^2+1$ is prime.