March 2024 Additive properties of the evil and odious numbers and similar sequences
Jean-Paul Allouche, Jeffrey Shallit
Funct. Approx. Comment. Math. 70(1): 55-69 (March 2024). DOI: 10.7169/facm/2108

Abstract

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.

Citation

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Jean-Paul Allouche. Jeffrey Shallit. "Additive properties of the evil and odious numbers and similar sequences." Funct. Approx. Comment. Math. 70 (1) 55 - 69, March 2024. https://doi.org/10.7169/facm/2108

Information

Published: March 2024
First available in Project Euclid: 15 March 2024

MathSciNet: MR4718478
Digital Object Identifier: 10.7169/facm/2108

Subjects:
Primary: 11B13
Secondary: 03D05 , 11B85 , 11L03 , 11P99 , 20F10 , 68Q45

Keywords: additive number theory , analytic number theory , automata theory , evil number , linear representation , number of representations , odious number , Trigonometric sum

Rights: Copyright © 2024 Adam Mickiewicz University

Vol.70 • No. 1 • March 2024
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