A family of digit functions with large periods

TL;DR

This paper explores a family of digit functions with large periods, proposing a hypothetical identity for differences in digit sums of multiples of odd integers, and provides verification algorithms and recursive formulas for these functions.

Contribution

It introduces a new hypothetical identity for digit sum differences, proves it for specific cases, and develops algorithms and recursions for general odd n.

Findings

Proved the identity for n=3 and n=5.

Empirically confirmed the identity for other odd n.

Derived recursive formulas depending on residue classes.

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.