  A Lucas sequence is a sequence of the general form $v_n = (\phi^n -
\bar{\phi}^n)/(\phi-\bar{\phi})$, where $\phi$ and $\bar{\phi}$ are real
algebraic integers such that $\phi+\bar{\phi}$ and $\phi\bar{\phi}$ are both
rational. Famous examples include the Fibonacci numbers, the Pell numbers, and
the Mersenne numbers. We study the monoid that is generated by such a sequence;
as it turns out, it is almost freely generated. We provide an asymptotic
formula for the number of positive integers $\leq x$ in this monoid, and also
prove Erd\H{o}s-Kac type theorems for the distribution of the number of
factors, with and without multiplicity. While the limiting distribution is
Gaussian if only distinct factors are counted, this is no longer the case when
multiplicities are taken into account.
