What is the maximum number of primes generated consecutively generated by a polynomial of degree $a$? Let $p(n)$ be a polynomial of degree $a$. Start of with plunging in arguments from zero and go up one integer at the time. Go on until you have come at an integer argument $n$ of which $p(n)$'s value is not prime and count the number of distinct primes your polynomial has generated. 

Question: what is the maximum number of distinct primes a polynomial of degree $a$ can generate by the process described above? Furthermore, what is the general form of such a polynomial $p(n)$?

This question was inspired by this article.
Thanks, 
Max
[Please note that your polynomial does not need to generate consecutive primes, only primes at consecutive positive integer arguments.]

Here is a related fact which might also be of interest.  There exists a polynomial in 26 variables with the property that, if you plug in integers for all 26 variables, and the output is a positive number (it will always be an integer), then the output is prime.  The polynomial can be found here:
http://en.wikipedia.org/wiki/Formula_for_primes
Furthermore, every prime occurs as an output of this polynomial.  This is a very indirect way to generate primes, because there is no easy way to know if a given input will give a positive output.
There is apparently another polynomial in 10 variables that does this, but Wikipedia doesn't explicitly write it.
