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 $17$.