We consider one-dimensional infinite chains of harmonic oscillators with stochastic perturbations and long-range interactions which have polynomial decay rate $|x|^{-θ}, x \to \infty, θ> 1$, where $x \in \mathbb{Z}$ is the interaction range. We prove that if $2< θ\le 3$, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent $\frac{3}{7-θ}$, while if $θ> 3$, then the exponent is $\frac{3}{4}$. The threshold is $ θ= 3$ because the derivative of the dispersion relation diverges as $k \to 0$ when $θ\le 3$.