Digit Functions of Integer Sequences

It was noticed by Benford [1] that the first nonzero digit in certain sets of real numbers is not uniformly distributed among the integers 1 through 9; in fact, the probability that this first, leftmost digit equals 3 is equal to log10(l + 3 " 1). He extended the analysis to the frequency of digits beyond the first for numbers obeying a particular probability law: the logarithmic distribution. This phenomenon of nonuniform distribution of digits has generated a considerable mathematical literature, particularly for the first digit, and has been shown to apply to the Fibonacci numbers [2], [3], [4], The purpose of this paper is to examine the probabilistic structure of the entire set of digits from certain integer sequences. The Fibonacci sequence provides one example. The essential results are that, for a large class of probability laws, the digits are not equiprobable and their values are correlated; but in the limit, as the ordinal number of the digits goes to infinity, the digit values approach equiprobability and their correlation goes to zero. However, under certain conditions, this limiting behavior does not occur; rather, the nonuniform behavior persists for all digits. In particular, subsequences of the Fibonacci sequence exist which exhibit "persistent Benford " behavior. Let 0) = {an} be a sequence of positive integers. Define a shifted sequence 0) of rationals an E Ub = [b " 1 9 1], for integer base b ^ 2, by where