One quadratic polynomial that merits consideration is

h(n) = n^2 + n + 41

It has the property that h(n) is prime for n=0..39.

Note that h(40) = 40(40 + 1) + 41.

Also, h(n) never has a factor smaller than 40 when n is an integer. I have a proof of this fact.

I put some more results on the web at

https://sites.google.com/site/mattc1anderson/home-1
I have some new results that I have not included on the internet.