A family of digit functions with large periods

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.

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