The phase behavior of selected alkane binary mixtures is studied using SAFT-VR, a version of the statistical associating fluid theory for potentials of variable attractive range (SAFT). We treat the n-alkane molecules as chains formed from united-atom hard-sphere segments with square-well potentials of variable range to describe the attractive interactions. We use a simple relationship between the number of carbon atoms in the n-alkane molecule and the number of segments in the united atom chains in order to predict the phase behavior of n-butane with other n-alkanes. The calculated vapor pressures and saturated liquid densities of the pure components are fitted to experimental data from the triple point to the critical point. These optimized parameters are rescaled by the respective experimental critical points and used to determine the critical lines and phase behavior of the mixtures. We use the Lorentz-Berthelot combining rule for the unlike interactions. We predict the phase behavior of n-butane + n-alkane binary mixtures, concentrating mainly on the critical region. The gas-liquid critical lines predicted by SAFT-VR for the n-alkane mixtures are in excellent agreement with the experimental data, and improve significantly on the results obtained with the simpler SAFT-HS approach where the attractive interactions are treated at the mean-field level.