15. Given $O A=1, O B=3, \overrightarrow{O A} \cdot \overrightarrow{O B}=0$, point $C$ is inside $\angle A O B$, and $\angle A O C=30^{\circ}$, set
$$
\overrightarrow{O C}=m \overrightarrow{O A}+n \overrightarrow{O B}(m 、 n \in \mathbf{R}) \text {. }
$$

Then $\frac{m}{n}=$ $\qquad$
15. $3 \sqrt{3}$.

Establish a Cartesian coordinate system. Without loss of generality, let
$$
\begin{array}{l}
A(1,0), B(0,3), C(\sqrt{3} y, y) . \\
\text { Then } \overrightarrow{O C}=(\sqrt{3} y, y)=m(1,0)+n(0,3) . \\
\text { Therefore }\left\{\begin{array}{l}
\sqrt{3} y=m, \\
y=3 n
\end{array} \Rightarrow \frac{m}{n}=3 \sqrt{3} .\right.
\end{array}
$$