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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.


marked as duplicate by Travis Willse, Matthew Towers, Robert Israel number-theory Nov 26 '18 at 14:26

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It's not number theory, but I've always found the Borwein integrals to be fascinating.



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