Skip to main content
SpringerLink
Log in

Positional number systems with digits forming an arithmetic progression

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

A novel digit system that arises in a natural way in a graph-theoretical problem is studied. It is defined by a set of positive digits forming an arithmetic progression and, necessarily, a complete residue system modulo the base b. Since this is not enough to guarantee existence of a digital representation, the most significant digit is allowed to come from an extended set. We provide explicit formulæ for the j th digit in such a representation as well as for the length. Furthermore, we study digit frequencies and average lengths, thus generalising classical results for the base-b representation. For this purpose, an appropriately adapted form of the Mellin-Perron approach is employed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Delange H.: Sur la fonction sommatoire de la fonction “somme des chiffres”. Enseignement Math. (2) 21, 31–47 (1975)

    MATH  MathSciNet  Google Scholar 

  2. Flajolet Ph., Grabner P., Kirschenhofer P., Prodinger H., Tichy R.F.: Mellin transforms and asymptotics: digital sums. Theor. Comput. Sci. 123, 291–314 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Grabner P.J., Kirschenhofer P., Prodinger H.: The sum-of-digits function for complex bases. J. Lond. Math. Soc. (2) 57(1), 20–40 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Graham R.L., Knuth D.E., Patashnik O.: Concrete Mathematics. A Foundation for Computer Science, 2nd edn. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  5. Heuberger C., Prodinger H.: On minimal expansions in redundant number systems: Algorithms and quantitative analysis. Computing 66, 377–393 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Heuberger C., Wagner S.G.: Maximizing the number of independent subsets over trees with bounded degree. J. Graph Theory 58, 49–68 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Knuth D.E.: Fundamental algorithms, 3rd edn. The Art of Computer Programming, vol. 1. Addison-Wesley, Reading (1997)

    Google Scholar 

  8. Okada T., Sekiguchi T., Shiota Y.: Applications of binomial measures to power sums of digital sums. J. Number Theory 52(2), 256–266 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Prodinger, H.: Digits and beyond, Mathematics and computer science, II (Versailles, 2002), Trends Math., Birkhäuser, Basel, pp. 355–377 (2002)

  10. Whittaker E.T., Watson G.N.: A course of modern analysis. Reprint of the 4th edn, 1927. Cambridge University Press, Cambridge (1963)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik B, Technische Universität Graz, Graz, Austria

    Clemens Heuberger

  2. Department of Mathematical Sciences, University of Stellenbosch, Matieland, South Africa

    Helmut Prodinger & Stephan G. Wagner

Authors
  1. Clemens Heuberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helmut Prodinger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephan G. Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clemens Heuberger.

Additional information

Part of this paper was written while C. Heuberger was a visitor at the Center of Experimental Mathematics at the University of Stellenbosch. He thanks the center for its hospitality. He is also supported by the Austrian Science Foundation FWF, project S9606, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory.”

H. Prodinger is supported by the NRF grant 2053748 of the South African National Research Foundation and by the Center of Experimental Mathematics of the University of Stellenbosch.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heuberger, C., Prodinger, H. & Wagner, S.G. Positional number systems with digits forming an arithmetic progression. Monatsh Math 155, 349–375 (2008). https://doi.org/10.1007/s00605-008-0008-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-008-0008-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation