4. The number of parabolas $n=$ $\qquad$ with vertex at integer points, directrix as the $y$-axis, and passing through the fixed point $A(2,0)$
4.4.

Let the vertex coordinates of the parabola be $(x, y)$.
Given that the y-axis is the directrix, the focus coordinates are $(2x, y)$.
Since point $A(2,0)$ lies on the parabola, combining the definition of the parabola, we get
$$
(2x-2)^{2}+y^{2}=4 \Rightarrow (x-1)^{2}+\frac{y^{2}}{4}=1 .
$$

This is the equation that the coordinates of the vertex of all parabolas satisfying the conditions must satisfy. The curve of the equation is an ellipse. Clearly, there are only four integer points on the ellipse: $(0,0)$, $(1,2)$, $(1,-2)$, $(2,0)$, and these are the four vertices of the ellipse. Therefore, there are 4 parabolas that meet the conditions. Among them, the parabola with the vertex at the origin $O$ is degenerate, i.e., the ray $Ox$. From a physics perspective, it can be considered the trajectory of a vertically upward projectile, which is, of course, a parabola.