Beyond odious and evil

J. P. Allouche, Benoit Cloitre, V. Shevelev

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


In a recent post on the Seqfan list the third author proposed conjectures concerning the summatory function of odious numbers (i.e., of numbers whose sum of binary digits is odd), and of 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.

Original languageEnglish
Pages (from-to)341-353
Number of pages13
JournalAequationes Mathematicae
Issue number2
StatePublished - 1 Apr 2016


  • Odious numbers
  • Thue–Morse sequence
  • evil numbers
  • iteration of sequences
  • summatory functions

ASJC Scopus subject areas

  • General Mathematics
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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