XXV OBM

Let $f(n)=n^2+5n+23$. For $n=-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4$ we get $f(n)=47,37,29,23,19,17,17,19,23,29,37,47,59$. Observe that $f(-n-5)=f(n)$, so this only involves calculating 7 terms. None of these are divisible by any of $2,3,5,7,11,13$. But all possible values mod $13$ are assumed by $f(n)$ for any 13 consecutive integers, similarly for 11, 7 etc. So $f(n)$ is not divisible by $2,3,5,7,11,13$ for any $n$. On the other hand $f(-2)=17$. So the answer is $17$.