  Let $\mathcal{A}$ be an alphabet of size $n\ge 2$. In this paper, we give a
complete description of primitive words $p\neq q$ over an alphabet
$\mathcal{A}$ of size $n\geq2$ such that $pq$ is non-primitive and $|p|=2|q|$.
In particular, if $l$ is s a positive integer, we count the cardinality of the
set $\mathcal{E}(l,\mathcal{A})$ of all couples $(p,q)$ of primitive words such
that $|p|=2|q|=2l$ and $pq$ is non-primitive. Then we give a combinatorial
formula for this cardinality and its asymptotic behavior, as $l$ or $n$ goes to
infinity.
