$13 \cdot 10$ On the coordinate plane, a point with both coordinates as integers is called an integer point. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$. Let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Find the value of $f(1)+f(2)+\cdots+f(1990)$.
(China High School Mathematics League, 1990)
[Solution] It is easy to see that the greatest common divisor $d=(n, n+3)$ of $n$ and $n+3$ is
$$
d=(n, \quad n+3)=\left\{\begin{array}{l}
3, \text { when } 3 \mid n \text {, } \\
1, \text { when } 3 \nmid n \text {, }
\end{array}\right.
$$

First, we prove that when $d=(n, n+3)=1$, there are no integer points on the line segment $O A_{n}$.
Otherwise, suppose $(m, l)$ is an integer point on $O A_{n}$, and satisfies $1 \leqslant m<n, 1 \leqslant$ $l<n+3$, then by

that is
$$
\begin{array}{c}
\frac{m}{l}=\frac{n}{n+3}, \\
m(n+3)=n l,
\end{array}
$$

and $(n, n+3)=1$, we know that $n \mid m$, which contradicts $m<n$.
When $d=(n, n+3)=3$, let $n=3 k$.
Then there are two integer points $(k, k+1),(2 k, 2 k+2)$ on the line segment $O A_{n}$.
Therefore, we have $\quad f(k)=\left\{\begin{array}{ll}2, & \text { when } 3 \mid k \text {, } \\ 0, & \text { when } 3 \nmid k \text {, }\end{array}\right.$
Thus, $f(1)+f(2)+\cdots+f(1990)=2 \cdot\left[\frac{1990}{3}\right]=1326$.