jmc

If $n$ is prime, then $g(n)=1$, so $n$ cannot divide $g(n)$. The primes less than or equal to $50$ are $$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47.$$There are $15$ of these primes. Also, if $n$ is the square of a prime, then $g(n)=\sqrt{n}$, so $n$ cannot divide $g(n)$. By looking at the list of primes we already generated, we see that there are four perfect squares of primes less than $50$. If $n$ is any other composite integer, then it can be decomposed into the product of integers $a$ and $b$ with both integers greater than $1$. We have that $ab$ divides $g(n)$ (since $g(n)$ is the product of a collection of integers including $a$ and $b$). Since $ab=n$, this implies that $n$ divides $g(n)$. As a result, there are $15+4=\boxed{19}$ values of $n$ for which $n$ does not divide $g(n)$.