  We study averages along the integers using the divisor function $d(n)$, and
defined as \begin{equation*}
  K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) , \end{equation*}
where $D(N) = \sum _{n=1} ^N d(n) $. We shall show that these averages satisfy
a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$, that is
\begin{equation*}
  \left( \frac{1}{N} \sum |K_Nf|^{p'} \right)^{1/p'}
  \lesssim
  \left(\frac{1}{N} \sum |f|^p \right)^{1/p} \end{equation*} as long as $f$ is
supported on $[0,N]$.
  We will also show that the associated maximal function $K^*f = \sup_N |K_N
f|$ satisfies $(p,p)$ sparse founds for $p \in (1,2)$, which implies that $K^*$
is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in
the Muckenhoupt $A_p$ class.
