  Given a real number $\lambda > 1$, we say that $d|n$ is a $\lambda$-middle
divisor of $n$ if $$ \sqrt{\frac{n}{\lambda}} < d \leq \sqrt{\lambda n}. $$ We
will prove that there are integers having an arbitrarily large number of
$\lambda$-middle divisors. Consider the word $$ \langle\! \langle n \rangle\!
\rangle_{\lambda} := w_1 w_2 ... w_k \in \{a,b\}^{\ast}, $$ given by $$ w_i :=
\left\{ \begin{array}{c l} a & \textrm{if } u_i \in D_n \backslash
\left(\lambda D_n\right), \\ b & \textrm{if } u_i \in \left(\lambda
D_n\right)\backslash D_n, \end{array} \right. $$ where $D_n$ is the set of
divisors of $n$, $\lambda D_n := \{\lambda d: \quad d \in D_n\}$ and $u_1, u_2,
..., u_k$ are the elements of the symmetric difference $D_n \triangle \lambda
D_n$ written in increasing order. We will prove that the language $$
\mathcal{L}_{\lambda} := \left\{\langle\! \langle n \rangle\! \rangle_{\lambda}
: \quad n \in \mathbb{Z}_{\geq 1} \right\} $$ contains Dyck words having an
arbitrarily large number of centered tunnels. We will show a connection between
both results.
