This question already has an answer here:
Some mathematical patterns stay true for a set of integers $1..n$ only to break at $n+1$.
What are some nontrivial examples where $n$ is ``large''?
As an example $x^2+x+41$ is prime for $x=1..40$, but not at $41$.
I am particularly looking for examples other than prime producing polynomials. Especially examples suitable for an introductory class.