A new approximate partition function is derived as a function of temperature and total number density of particles in the given system, and three adjustable parameters. The derivation assumes that we can simulate the calculations of the partition function for hydrogen by means of averages of the energies and sums of the statistical weights.We present the procedure and mathematical process to obtain an approximate analytic function and its derivatives that depend on those parameters. The comparisons with other calculations reported in the literature show good agreement. The free parameters of this function are calculated and given in a table for all the ions of the first 20 atomic species.