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Beyond odious and evil

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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.

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Authors and Affiliations

  1. CNRS, Institut de Math. de Jussieu, Université P. et M. Curie, Case 247, 4 Place Jussieu, 75252, Paris Cedex 05, France

    J.-P. Allouche

  2. 19, rue Louise Michel, 92300, Levallois-Perret, France

    Benoit Cloitre

  3. Department of Mathematics, Ben-Gurion University of the Negev, Beersheva, Israel

    V. Shevelev

Authors
  1. J.-P. Allouche
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  2. Benoit Cloitre
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  3. V. Shevelev
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Corresponding author

Correspondence to J.-P. Allouche.

Additional information

The author was partially supported by the ANR project “FAN” (Fractals et Numération).

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Allouche, JP., Cloitre, B. & Shevelev, V. Beyond odious and evil. Aequat. Math. 90, 341–353 (2016). https://doi.org/10.1007/s00010-015-0345-3

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  • DOI: https://doi.org/10.1007/s00010-015-0345-3

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