The phase diagrams of 4He adsorbed on the external surfaces of single armchair carbon nanotubes with radii in the range 3.42–10.85 Å are calculated using the diffusion Monte Carlo method. For nanotubes narrower than a (10,10) one, the ground state is an incommensurate solid similar to the one found for H2 on the same substrates. For wider nanotubes, the phase with the minimum energy per particle is a liquid layer. Curved √3×√3 registered solids similar to the ones found on graphene and graphite were unstable for all the tubes considered.