Aequationes mathematicae

, Volume 90, Issue 2, pp 341–353 | Cite as

Beyond odious and evil

  • J.-P. Allouche
  • Benoit Cloitre
  • V. Shevelev


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.


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

Mathematics Subject Classification

11A63 11B83 11B85 11A07 05A15 68R15 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.CNRS, Institut de Math. de JussieuUniversité P. et M. Curie, Case 247Paris Cedex 05France
  2. 2.Levallois-PerretFrance
  3. 3.Department of MathematicsBen-Gurion University of the NegevBeershevaIsrael

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