  Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the
relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell_u(m)=lcm(m, z_u(m))$, for
$(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove
that $$\sum_{\substack{m>x\\ (m,a_2)=1}}\frac{1}{\ell_u(m)}\leq
\exp(-(1/\sqrt{6}-\varepsilon+o(1))\sqrt{(\log x)(\log \log x)}),$$ when $x$ is
sufficiently large in terms of $\varepsilon$, and where the $o(1)$ depends on
$u$. Moreover, if $g_u(n)=\gcd(n,u_n)$, we will show that for every $k\geq 1$,
$$\sum_{n\leq x}g_u(n)^{k}\leq x^{k+1}\exp(-(1+o(1))\sqrt{(\log x)(\log \log
x)}),$$ when $x$ is sufficiently large and where the $o(1)$ depends on $u$ and
$k$. This gives a partial answer to a question posed by C. Sanna. As a
by-product, we derive bounds on $#\{n\leq x: (n, u_n)>y\}$, at least in certain
ranges of $y$, which strengthens what already obtained by Sanna. Finally, we
start the study of the multiplicative analogous of $\ell_u(m)$, finding
interesting results.
