Self-consistent Monte Carlo simulations of the positive column of gas discharges with a range of gas densities and currents are described. These simulations are performed using highly efficient Monte Carlo codes for electron transport in cylindrical geometry and using a realistic particle model of ion transport. No internal boundary conditions or assumptions about the plasma-sheath boundary are used in the simulations. Techniques to speed up the convergence are discussed. The self-consistent radial potential for each plasma is presented as a Taylor series expansion in radius squared terms. The 16 converged simulations presented here were checked using two independent codes, written by different authors, in different languages and executed on different computers. These positive column results are useful in comparisons with other types of fluid and kinetic theory simulations.