  Given three pairwise coprime positive integers $a_1,a_2,a_3 \in \mathbb{Z}^+$
we show the existence of a relation between the sets of the first elements of
the three quotients $\frac{\langle a_i,a_j \rangle}{a_k}$ that can be made for
every $\{i.j,k\}=\{1,2,3\}$. Then we use this result to give an improved
version of Johnson's semi-explicit formula for the Frobenius number
$g(a_1,a_2,a_3)$ without restriction on the choice of $a_1,a_2,a_3$ and to give
an explicit formula for a particular class of numerical semigroups.
