Distribution of Leading Digits of Numbers

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