jmc

The potential square factors are $4$, $9$, $16$, $25$, $36$, and $49$. $4$ divides $12$ of the numbers. $9$ divides $5$ of the numbers, but we've counted $4\cdot 9=36$ twice, so we subtract $1$. $16$ divides $3$ of the numbers, but each of those is also divisible by $4$, so we don't count them. $25$ divides $2$. $36$ divides $1$, itself, but it's already been counted. Finally, $49$ divides $1$. Thus, our final answer is $12+(5-1)+2+1=\boxed{19}$.