Beyond odious and evil

TL;DR

This paper proves conjectures about the summatory functions of odious and evil numbers, explores their generalized sequences, and characterizes the classical sequences through functional equations.

Contribution

It provides proofs for conjectures on odious and evil numbers and introduces a new characterization via functional equations for these sequences.

Findings

Proof of conjectures on odious and evil numbers' summatory functions

Characterization of odious and evil sequences through functional equations

Analysis of generalized odious and evil number sequences

Abstract

In a recent post on the Seqfan list the third author proposed a conjecture concerning the summatory function of odious numbers (i.e., of numbers whose sum of binary digits is odd), and its analog for evil numbers (i.e., of numbers whose sum of binary digits is even). We prove these conjectures here. We will also study the sequences of "generalized" odious and evil numbers, and their iterations, giving in particular a characterization of the sequences of usual odious and evil numbers in terms of functional equations satisfied by their compositions.