Growth rate of primes of the form $a n+1$, with $n$ fixed If we fix a natural $n$, and let $a$ vary, how big can $a n + 1$ be before the first prime is encountered, in terms of $n$?  In other words, let $n = 32$.  Then, we have:
$$1(32) + 1 = 33 = 3 * 11 \text{ is composite}$$
$$2(32) + 1 = 65 = 5 * 13 \text{ is composite}$$
$$3(32) + 1 = 97 \text{ is prime}$$
The gist of the question is to try to find asymptotic bounds on $a$.  I believe that $a \in O(\log{(n)})$, and therefor $an+1 < c \cdot (n \log{(n)})$ for some $c$, but I'm not sure of this.  If this isn't the case, could someone please provide the correct bounds.

Let $ak+b$ be an arithmetic progression, $k = 1,2,3,\ldots$. Dirichlet theorem of primes in arithmetic progression says that number of prime $\le x$ which are of the form $ak+b$ is
$$
\pi_a(x) \approx \frac{\pi(x)}{\varphi(a)} \approx \frac{x}{\varphi(a)\log x}
$$
where $\varphi(n)$ is the Euler totient function. Hence heuristically, to get the first prime, we must have
$$
\frac{x}{\varphi(a)\log x} \ge 1
$$
or 
$$
\frac{x}{\log x} \ge \varphi(a)
$$
which can be solved for $x$ for any given $a$. 
Here is an approximate solution. Let $p_{\varphi(a)}$ be the $\varphi(a)$-th prime. Then, the smallest $x$ satisfying the above inequality is about $p_{\varphi(a)}$ which is about
$$
p_{\varphi(a)} \approx \varphi(a)\log \varphi(a) + \varphi(a)\log \log \varphi(a)
- \varphi(a)
$$
Then on a average the minimum value of $k$ is about 
$$
\frac{\varphi(a)\log \varphi(a) + \varphi(a)\log \log \varphi(a)
- \varphi(a)}{a}
$$
