smc

First, note that $n$ does not have a prime number larger than $61$ as one of its factors. Also, note that $n$ does not equal $1$. Therefore, since the prime factorization of $n$ only has primes from $2$ to $59$, $n$ and $P$ share at least one common factor other than $1$. Therefore $P+n$ is not prime for any $n$, so the answer is $\Rightarrow \boxed{0}$.