Proof for factors of a number
If a number $n$ has a factor $m$ in the range $n/4 \leq m \leq n/2$, then the quotient $k = n/m$ satisfies $2 \leq k \leq 4$, that is, $k \in \{2,3,4\}$. This should give you a fast algorithm for determining whether such a factor $m$ exists.