I've seen that, a possible defining condition for the reciprocal lattice is:
$\vec R_s \cdot \vec G=2 \pi l$, where, $R_s= n_1 \vec a_1+n_2 \vec a_2+n_3 \vec a_3$ is the direct lattice, $\vec G$ is the reciprocal lattice, and $l \in \mathbb{Z}$
Using this definition in the one dimensional case, i find a wrong result, indeed: let $a$ be the distance between atoms, it follows that $R_s=na$ is the direct lattice. The defining equation, for the reciprocal lattice, is $R_sG=2 \pi l$. Substituting it becomes $naG=2 \pi l$ and so $G=\frac {2 \pi l}{a n}$, which is wrong, because it should be $G=\frac {2 \pi l}{a }$. Why do i have that extra integer, $n$, at the denominator?