TITLE: Is the notorious $n^2 + n + 41$ prime generator the last of its type?
QUESTION [92 upvotes]: The polynomial $n^2+n+41$ famously takes prime values for all $0\le n\lt 40$. I have read that this is closely related to the fact that 163 is a Heegner number, although I don't understand the argument, except that the discriminant of $n^2+n+41$ is $-163$. The next smaller Heegner number is 67, and indeed $n^2+n+17$ shows the same behavior, taking prime values for $0\le n\lt 16$.  
But 163 is the largest Heegner number, which suggests that this is the last such coincidence, and that there might not be any $k>41$ such that $n^2+n+k$ takes on an unbroken sequence of prime values.
Is this indeed the case?  If not, why not?
Secondarily, is there a $k>41$, perhaps extremely large, such that $n^2+n+k$ takes on prime values for $0<n<j$ for some $j>39$?

REPLY [3 votes]: Probably, it's not the last one. I just discovered that some similar formulas, such as n^2 - n + 55661, actually contain more primes than n^2 - n + 41, tested with n=1 up to 10^7.
Here are some examples:
41        -> 2208197 primes
55661     -> 2478942 primes
333491    -> 2535780 primes
701147    -> 2587904 primes

Other similar formulas include:
n^2 - n + 361637
n^2 - n + 383681
n^2 - n + 601037
n^2 - n + 206807