10. In the coordinate plane, a point whose both coordinates are integers is called an integer point. For any positive integer $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$, and let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Then $f(1)+f(2)+\cdots+f(1990)=$ $\qquad$ .
According to the problem, $y=\frac{n+3}{n} x=x+\frac{3 x}{n} \in \mathbf{N}^{*}, x=1,2, \cdots, n-1$, thus $n \mid 3 x$. Since $n \nmid x \Rightarrow n=3 k \Rightarrow k \mid x$, hence $x=k, 2 k$. That is, when $n=3 k$, $f(n)=2$; when $n \neq 3 k$, $f(n)=0$.
Therefore, $f(1)+f(2)+\cdots+f(1990)=2 \times 663=1326$.