  In [CaballeroHooleyDelta], we associated a Dyck word $\langle\! \langle n
\rangle\! \rangle_{\lambda}$ to any pair $(n, \lambda)$ consisting of an
integer $n \geq 1$ and a real number $\lambda > 1$. The goal of the present
paper is to show a relationship between the factorization of $\langle\! \langle
n \rangle\! \rangle_{\lambda}$ as the concatenation of irreducible Dyck words
and the distribution of the divisors of $n$. In particular, we will provide a
characterization of $\lambda$-densely divisible numbers (these numbers were
introduced in [castryck2014new]).
