AHSME 1960 Problem 33

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times\ldots \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4,\ldots, 59$. Let $N$ be the number of primes appearing in this sequence. Then $N$ is: $\textbf{(A)}\ 0 \textbf{(B)}\ 16 \textbf{(C)}\ 17 \textbf{(D)}\ 57 \textbf{(E)}\ 58$