Advanced Number Theory · Challenge problem

For a positive integer $n$, let $z(n)$ be the number of trailing zeros of $n!$ when written in base $10$. For example $z(10) = 2$ and $z(24) = 4$, and no factorial ends in exactly $5$ zeros. How many integers $k$ with $1 \le k \le 1000$ are not equal to $z(n)$ for any positive integer $n$?