A family of digit functions with large periods

Abstract

For odd n ≥ 3, we consider a general hypothetical identity for the differences S n, 0 (x) of multiples of n with even and odd digit sums in the base n − 1 in interval [0, x), which we prove in the cases n = 3 and n = 5 and empirically confirm for some other n. We give a verification algorithm for this identity for any odd n. The hypothetical identity allows to give a general recursion for S n, 0 (x) for every integer x depending on the residue of x modulo p(n) = 2n(n − 1) n−1 , such that p(3) = 24, p(5) = 2560, p(7) = 653184, etc.