Jean-Paul Allouche, Jeffrey Shallit

Funct. Approx. Comment. Math. Advance Publication, 1-15, (2023) DOI: 10.7169/facm/2108
KEYWORDS: additive number theory, number of representations, odious number, evil number, automata theory, analytic number theory, Trigonometric sum, linear representation, 11B13, 11B85, 11P99, 03D05, 68Q45, 20F10, 11L03

First we reprove two results in additive number theorydue to Dombi and Chen & Wang, respectively, on the number ofrepresentations 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 aboutthe 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 $\pm 1$ characteristic sequences of these integers.