Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and , n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.