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.
It's not number theory, but I've always found the Borwein integrals to be fascinating.
https://en.wikipedia.org/wiki/Borwein_integral