  We perform a numerical study of transport properties of a one-dimensional
chain with couplings decaying as an inverse power $r^{-(1+\sigma)}$ of the
intersite distance $r$ and open boundary conditions, interacting with two heat
reservoirs. Despite its simplicity, the model displays highly nontrivial
features in the strong long-range regime, $-1<\sigma<0$. At weak coupling with
the reservoirs, the energy flux departs from the predictions of perturbative
theory and displays anomalous superdiffusive scaling of the heat current with
the chain size. We trace back this behavior to the transmission spectrum of the
chain, which displays a self-similar structure with a characteristic
sigma-dependent fractal dimension.
