Resonance

, Volume 19, Issue 11, pp 1038–1046 | Cite as

Some remarks on iterated maps of natural numbers

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Abstract

The iterates of maps f: ℕ → ℕ given as a function of the digits of the number written in a fixed base b are dealt with here. For such maps, the iterates end up in a finite collection of cycles. The number and length of such cycles have arithmetic significance.

Keywords

Iterates of maps integer sequences Steinhaus problem 
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Suggested Reading

  1. [1]
    H Steinhaus, One Hundred Problems in Elementary Mathematics, Dover Publications, 1979.Google Scholar
  2. [2]
    R K Guy, Unsolved Problems in Number Theory, Springer-Verlag, New Yor, 2nd edition, 2004.CrossRefGoogle Scholar
  3. [3]
    E El-Sedy and S Siksek, On happy numbers, Rocky Mountain J. Math., Vol.30, No.2, pp.565–570, 2000.CrossRefGoogle Scholar
  4. [4]
    H Pan, On consecutive happy numbers, Journal of Number Theory, Vol.128, No.6, pp.1646–1654, 2008.CrossRefGoogle Scholar
  5. [5]
    H G Grundman and E A Teeple, Generalized happy numbers. Fibonacci Quart., Vol.39, No.5, pp.462–466, 2001.Google Scholar
  6. [6]
    H G Grundman and E A Teeple, Heights of happy numbers and cubic happy numbers, Fibonacci Quart., Vol.41, 2003, No.4, pp.301–306, 2003.Google Scholar
  7. [7]
    H G Grundman and E A Teeple, Sequences of generalized happy numbers with small bases, J. Integer Seq., Vol.10, No.1, Article 07.1.8,6, 2007.Google Scholar
  8. [8]
    H G Grundman and E A Teeple, Sequences of consecutive happy numbers, Rocky Mountain J. Math., Vol.37, No.6, pp.1905–1916, 2007.CrossRefGoogle Scholar
  9. [9]
    H G Grundman and E A Teeple, Iterated sums of fifth powers of digits, Rocky Mountain J. Math., Volo.38, No.4, pp.1139–1146, 2008.CrossRefGoogle Scholar
  10. [10]
    A F Beardon, Sums of squares of digits, Math. Gazette, Vol.82, pp.379–388. 1998.CrossRefGoogle Scholar
  11. [11]
    K Iseki, A problem of number theory, Proc. Japan Acad., Vol.36, pp.578–583, 1960.CrossRefGoogle Scholar
  12. [12]
    K Iseki, Necessary results for computation of cyclic parts in Steinhaus problem, Proc. Japan Acad., Vol.36, pp.650–651, 1960.CrossRefGoogle Scholar
  13. [13]
    S P Mohanty and H Kumar, Powers of sums of digits, Math. Mag., Vol.52, No.5, pp.310–312, 1979.CrossRefGoogle Scholar
  14. [14]
    K Chikawa and K Iseki, Unsolved problems in number theory. Steinhaus Problem, Proc. Japan Acad., Vol.38, p.271, 1962.CrossRefGoogle Scholar
  15. [15]
    J Gilmer, On the density of happy numbers, preprint, 2011, posted at: http://arxiv.org/pdf/1110.3836v3.pdfGoogle Scholar

Copyright information

© Indian Academy of Sciences 2014

Authors and Affiliations

  1. 1.Department of MathematicsQueen’s UniversityKingstonCanada

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