Additive Properties of the Evil and Odious Numbers and Similar Sequences

TL;DR

This paper reestablishes known additive properties of evil and odious numbers using automata theory, introduces new results on sums involving these numbers, and extends analysis to similar sequences with advanced number theory techniques.

Contribution

It combines automata theory and analytic number theory to prove new additive properties of evil, odious, and similar sequences, extending previous results.

Findings

Reproved known results on representations as sums of evil and odious numbers.

Proved new results on numbers represented by five summands.

Analyzed tenfold sums and k-fold sums of related sequences.

Abstract

First we reprove two results in additive number theory due to Dombi and Chen & Wang, respectively, on the number of representations of n as the sum of two odious or evil numbers, using techniques from automata theory and logic. We also use this technique to prove a new result about the numbers represented by five summands. Furthermore, we prove some new results on the tenfold sums of the evil and odious numbers, as well as k-fold sums of similar sequences of integers, by using techniques of analytic number theory involving trigonometric sums associated with the (+-1)-characteristic sequences of these integers.