In 2D your problem is related to the Farey sequence.
In fact the geometrical rendering of this sequence is as sketched below.
At level $7$ for example, you have all the ratios already contained in precedent levels (the small black points) plus
the newly added (big red points) $1/7, 2/7, \cdots, 6/7$.
In the Wikipedia article above you find all the explanation of how to count the number of terms in a Farey sequence of a certain level,
as well the difference with the previous levels, and I am not going to repeat it here.
As a matter of fact, in 2D a segment from the origin to the point$(X,Y)$ (assume wlog $0 \le Y \le X$) will not intercept an intermediate point with integral coordinates
if the fraction $Y/X$ is irreducible, i.e. if $\gcd (Y,X) =1$ , otherwise if $1 < \gcd (Y,X) =g$ then $(X,Y) = g (X/g, Y/g)$.
Also consider that $\gcd(Y,X) =\gcd(Y, X-Y)$, that is we can tilt the triangle above to have the right corner at the origin and count the points by diagonals $X+Y=n$.
Similarly in 3D, a segment from the origin to the point $(X,Y,Z)$ will not intercept any other point before if $\gcd(X,Y,Z)=1$, otherwise
we will have $(X,Y,Z)= g (X/g, Y/g, Z/g)$.
Same for higher dimensions, and again we can count the points by diagonal planes
$$
\eqalign{
& \left\{ {\left( {X_1 ,X_2 , \cdots ,X_m } \right)\;:\;\gcd \left( {X_1 ,X_2 , \cdots ,X_m } \right) = g\; \wedge \;\;0 \le X_1 \le X_2 \le \cdots \le X_m \le L} \right\} = \cr
& = \left\{ {\left( {X_1 ,X_2 , \cdots ,X_m } \right)\;:\;\gcd \left( {X_1 ,X_2 , \cdots ,X_m } \right) = g\; \wedge \;\;0 \le X_1 + X_2 + \cdots + X_m \le L} \right\} \cr}
$$
The problem goes under the standard denomination of visible lattice points.
At the same time we can consider
$$
\left\{ {\left( {X_1 ,X_2 , \cdots ,X_m } \right)\;:\;\quad \gcd \left( {X_1 ,X_2 , \cdots ,X_m } \right) = 1\; \wedge \;\;X_1 + X_2 + \cdots + X_m = n} \right\}
$$
as the set of the Compositions of $n$ into $m$ relatively prime parts,
the compositions being understood in the "weak" sense if $0 \le X_j$ and in the standard sense with $1 \le X_j$ .
Under this perspective, take a look at this interesting work by H.W. Gould
"Binomial Coefficients, the Bracket Function, and Compositions with Relatively Prime Summands".
Therein, having denoted by $ R_{\,m} (n)$ the number of standard compositions as above, i.e:
$$
R_{\,m} (n) = \sum\limits_{\left\{ {\matrix{
{1\, \le \,X_k } \cr
{\gcd \left( {X_1 ,X_2 , \cdots ,X_m } \right) = 1} \cr
{X_1 + X_2 + \cdots + X_m = n} \cr
} } \right.} 1
$$
it is demonstrated that
$$ \bbox[lightyellow] {
R_{\,m} (n) = \sum\limits_{d\backslash n} {\left( \matrix{
d - 1 \cr
m - 1 \cr} \right)\mu \left( {n/d} \right)}
}$$
where $\mu$ denotes the Möbius function.
Example:
in 2D, the formula above gives
$$
R_{\,2} (n)\quad \left| {\;2 \le n} \right.\;\; = \varphi (n)
= {\rm 1}{\rm , 2}{\rm , 2}{\rm , 4}{\rm , 2}{\rm , 6}{\rm , 4}{\rm , 6}{\rm , 4}{\rm ,} \cdots
$$
in 3D, we get
$$
R_{\,3} (n)\quad \left| {\;3 \le n} \right.\;\; =
{\rm 1}{\rm , 3}{\rm , 6}{\rm , 9}{\rm , 15}{\rm , 18}{\rm , 27}{\rm , 30}{\rm , 45}{\rm , 42}{\rm , 66}{\rm , 63}{\rm , 84}{\rm , 84}
{\rm , 120}{\rm , 99}
$$
which is OEIS A000741
in 4D it is OEIS A000742
