$P_1, P_2, P_3, \ldots$ are points on line $y=x$.
Let
$$
\begin{gathered}
O P_1=1 \\
P_1\left(x_1, x_1\right)
\end{gathered}
$$


$$
\begin{aligned}
& \text { But } \quad O P_1=1 \quad \Rightarrow \sqrt{x_1^2+y_1^2}=1 \\
& \Rightarrow \quad \sqrt{x_1^2+x_1^2}=1 \quad \Rightarrow \quad 2 x_1^2=1 \\
& \Rightarrow \quad x_1^2=\frac{1}{2} \quad \Rightarrow \quad x_1=\frac{1}{\sqrt{2}} \\
&
\end{aligned}
$$

So, $P_1\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) \quad O P_2=2 O P_1=2 \times 1$
$$
\begin{aligned}
& \sqrt{x_2^2+x_2^2}=2 \quad 2 x_2^2=4 \\
\Rightarrow \quad & x_2^2=2 \Rightarrow x_2=\sqrt{2}
\end{aligned}
$$

So, $P_2(\sqrt{2}, \sqrt{2})$
Similarly, $\mathrm{OP}_3=3 \mathrm{OP}_2=6$
$$
2 x_3^2=6^2 \Rightarrow x_3=3 \sqrt{2}
$$

Continuing this way, we get
$$
\begin{array}{lc}
& O P_7=7 \times O P_6=7 \times 720 \\
\Rightarrow & \sqrt{2 x_7^2}=5040 \\
\Rightarrow & x_7^2=\frac{5040 \times 5040}{2} \\
\Rightarrow & x_7=2520 \sqrt{2} \\
\text { So, } & P_7=(2520 \sqrt{2}, 2520 \sqrt{2})
\end{array}
$$

By comparing with $P_n=(2520 \sqrt{2}, 2520 \sqrt{2})$, we get $n=7$