Advertisement

Aequationes mathematicae

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

Beyond odious and evil

  • J.-P. Allouche
  • Benoit Cloitre
  • V. Shevelev
Article
  • 88 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

11A63 11B83 11B85 11A07 05A15 68R15 

References

  1. 1.
    Allouche J.-P., Rampersad N., Shallit J.: On integer sequences whose first iterates are linear. Aequationes Math. 69, 114–127 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Allouche, J.-P., Shallit, J.: The ubiquitous Prouhet–Thue–Morse sequence. In: Sequences and their Applications (Singapore, 1998). Springer Ser. Discrete Math. Theor. Comput. Sci., pp. 1–16. Springer, London (1999)Google Scholar
  3. 3.
    Berlekamp, E., Conway, J., Guy, R.: Winning Ways for Your Mathematical Plays, vols. 1 and 2. Academic Press, London (1982)Google Scholar
  4. 4.
    Berlekamp, E., Conway, J., Guy, R.: Winning Ways for Your Mathematical Plays, vol. 3, 2nd edn. A. K. Peters, Natick (2003)Google Scholar
  5. 5.
    Cloitre, B., Sloane, N.J.A., Vandermast, M.J.: Numerical analogues of Aronson’s sequence. J. Integer Seq. 6 (2003) Art. 03.2.2Google Scholar
  6. 6.
    Emmer, M.: Numeri malefici (evil numbers): homage to Fabio Mauri. In: Emmer, M. (ed.) Imagine Math 2: Between Culture and Mathematics, pp. 139–150. Springer, Italia (2013)Google Scholar
  7. 7.
    Fine N.J.: The distribution of the sum of digits (mod p), Bull. Am. Math. Soc. 71, 651–652 (1965)CrossRefMATHGoogle Scholar
  8. 8.
    Gel’fond A.O.: Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith. 13, 259–265 (1967/1968)Google Scholar
  9. 9.
    Guy, R.K.: Impartial games. In: Combinatorial Games, vol. 29. MSRI Publications (1995). http://library.msri.org/books/Book29/files/imp. Accessed 18 Feb 2015
  10. 10.
    Laohakosol V., Yuttanan B.: Iterates of increasing sequences of positive integers. Aequationes Math. 87, 89–103 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Nietzche, F.W.: Beyond Good and Evil: prelude to a Philosophy of the Future, Edited by Rolf-Peter Horstmann, Edited and translated by Judith Norman. Cambridge Texts in the History of Philosophy, Cambridge (2002)Google Scholar
  12. 12.
    On-Line Encyclopedia of Integer Sequences, available electronically at the URL http://oeis.org
  13. 13.
    Sarkaria K.S.: Roots of translations. Aequationes Math. 75, 304–307 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Targoński, G.: Topics in iteration theory, In: Studia Mathematica: Skript, vol. 6. Vandenhoeck & Ruprecht, Göttingen (1981)Google Scholar
  15. 15.
    Wolfram Mathworld. http://mathworld.wolfram.com/EvilNumber.html. Accessed 18 Feb 2015

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

Personalised recommendations