Floor Times Fractional Part

For a real number $t$, $\lfloor t \rfloor$ is the greatest integer that is less than or equal to $t$, while $\{t\}$ is the fractional part of $t$. For how many real numbers $0 < a < 20$ is $\lfloor a \rfloor \cdot \{a\} = n^2$ for some positive integer $n$?