  In the present work we study the Fermi--Pasta--Ulam (FPU) $\beta $--model
involving long--range interactions (LRI) in both the quadratic and quartic
potentials, by introducing two independent exponents $\alpha_1$ and $\alpha_2$
respectively, which make the {forces decay} with distance $r$. Our results
demonstrate that weak chaos, in the sense of decreasing Lyapunov exponents, and
$q$--Gaussian probability density functions (pdfs) of sums of the momenta,
occurs only when long--range interactions are included in the quartic part.
More importantly, for $0\leq \alpha_2<1$, we obtain extrapolated values for $q
\equiv q_\infty >1$, as $N\rightarrow \infty$, suggesting that these pdfs
persist in that limit. On the other hand, when long--range interactions are
imposed only on the quadratic part, strong chaos and purely Gaussian pdfs are
always obtained for the momenta. We have also focused on similar pdfs for the
particle energies and have obtained $q_E$-exponentials (with $q_E>1$) when the
quartic-term interactions are long--ranged, otherwise we get the standard
Boltzmann-Gibbs weight, with $q=1$. The values of $q_E$ coincide, within small
discrepancies, with the values of $q$ obtained by the momentum distributions.
