We consider heat transport in one-dimensional harmonic chains attached at its ends to Langevin heat baths. The harmonic chain has mass impurities where the separation $d$ between any two successive impurities is randomly distributed according to a power-law distribution $P(d)sim 1/d^{alpha+1}$, being $alpha>0$. In the regime where the first moment of the distribution is well defined ($1<alpha<2$) the thermal conductivity $kappa$ scales with the system size $N$ as $kappasim N^{(alpha-3)/alpha}$ for fixed boundary conditions, whereas for free boundary conditions $kappasim N^{(alpha-1)/alpha}$ if $Ngg1$. When $alpha=2$, the inverse localization length $lambda$ scales with the frequency $omega$ as $lambdasim omega^2 ln omega$ in the low frequency regime, due to the logarithmic correction, the size scaling law of the thermal conductivity acquires a non-closed form. When $alpha>2$, the thermal conductivity scales as in the uncorrelated disorder case. The situation $alpha<1$ is only analyzed numerically, where $lambda(omega)sim omega^{2-alpha}$ which leads to the following asymptotic thermal conductivity: $kappa sim N^{-(alpha+1)/(2-alpha)}$ for fixed boundary conditions and $kappa sim N^{(1-alpha)/(2-alpha)}$ for free boundary conditions.
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