Questions about primes made from consecutive numbers starting from 1

Let $b_n=\overline{a_1a_2a_3\dots a_n}$ and $a_n=n$. For example $b_{11}= 1234567891011$. I have a couple of questions about the primes in the sequence $b_n$.

1. What is the lowest number $n$ such that $b_n$ is prime (if there is is an $n$)?
2. Are there infintely many primes in the sequence $b_n$?
3. Do these primes have a name?

The first question was an excercise in Clifford Pickover's A Passion For Mathematics. I don't know if there is a way to find the answer to the first question without using a computer.

Like in Does there exist a prime that is only consecutive digits starting from 1?, $n$ cannot be an even number (or else $b_n$ would be even). Thanks.

• If noone finds an answer, the recursion is $$b_{n+1} = 10^{\lfloor \log_{10}(n+1) \rfloor} b_n + n + 1$$ So you can write a script to test, similar to the linked question. – AlexR Sep 30 '14 at 7:15
• @AlexR Thanks, I still hope someone finds an answer though. – Joao Sep 30 '14 at 7:16
• You should add an example just to make sure people understand the notation. For example, I believe that $b_{11}$ is the thirteen-digit number 1,234,567,891,011; confirming that would be helpful. – Greg Martin Sep 30 '14 at 7:23
• None of $b_2,\dots,b_{2000}$ is prime. But a probabilistic heuristic suggests that infinitely many of the $b_n$ should be prime, although they will be extremely rare. – Greg Martin Sep 30 '14 at 7:23