title: A first digit theorem for powerful integer powers.
abstract: For any fixed power exponent, it is shown that the first digits of powerful integer powers follow a generalized Benford law (GBL) with size-dependent exponent that converges asymptotically to a GBL with the inverse double power exponent. In particular, asymptotically as the power goes to infinity these sequences obey Benford's law. Moreover, the existence of a one-parametric size-dependent exponent function that converges to these GBL's is established, and an optimal value that minimizes its deviation to two minimum estimators of the size-dependent exponent is determined. The latter is undertaken over the finite range of powerful integer powers less than [Formula: see text], where [Formula: see text] is a fixed power exponent.
authors.affiliation: Swiss Mathematical Society, University of Fribourg, 1700 Fribourg, Switzerland.
authors.forename: Werner
authors.initials: W
authors.lastname: Hürlimann
keywords: Asymptotic counting function; First digit; Mean absolute deviation; Powerful number; Probabilistic number theory; Probability weighted least squares