Composition of happy functions

Passawan Noppakeaw, Niphawan Phoopha and Prapanpong Pongsriiam
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 3, Pages 13-20
DOI: 10.7546/nntdm.2019.25.3.13-20
Full paper (PDF, 193 Kb)

Details

Authors and affiliations

Passawan Noppakeaw
Department of Mathematics, Faculty of Science
Silpakorn University, Nakhon Pathom, 73000, Thailand

Niphawan Phoopha
Department of Mathematics, Faculty of Science
Silpakorn University, Nakhon Pathom, 73000, Thailand

Prapanpong Pongsriiam
Department of Mathematics, Faculty of Science
Silpakorn University, Nakhon Pathom, 73000, Thailand

Abstract

For positive integers e\geq 1 and b\geq 2, let S_{e,b}:\mathbb{N}\to\mathbb{N} be defined by

    \[S_{e,b}(x)=a^e_k+a^e_{k-1}+\cdots +a^e_1\]

if x = (a_ka_{k-1}\ddots a_1)_{b} = a_kb^{k-1}+a_{k-1}b^{k-2}+\cdots+a_2b+a_1 is the expansion of x in base b. We call S_{e,b} an (e,b)-happy function. Let g be a composition of various (e,b)-happy functions. We show that, for any given x\in\mathbb{N}, the iteration sequence (g^{(n)}(x))_{n\geq 0} either converges to a fixed point or eventually becomes a cycle. Here g^{(0)} is the identity function mapping x to x for all x and g^{(n)} is the n-fold composition of g. In addition, we prove that the number of all possible fixed points and cycles is finite. Examples are also given.

Keywords

  • Happy number
  • Happy function
  • Digit
  • Dynamic
  • Iteration

2010 Mathematics Subject Classification

  • 11A63
  • 26A18

References

  1. El-Sedy, E. & Siksek, S. (2000). On happy numbers, The Rocky Mountain Journal of Mathematics, 30, 565–570.
  2. Gilmer, J. (2011). On the density of happy numbers, posted at: http://arxiv.org/pdf/1110.3836v3.pdf
  3. Grundman, H. G. & Teeple, E. A. (2001). Generalized happy numbers, The Fibonacci Quarterly, 39, 462–466.
  4. Grundman, H. G. & Teeple, E. A. (2008). Iterated sums of fifth powers of digits, The Rocky Mountain Journal of Mathematics, 38, 1139–1146.
  5. Guy, R. K. (2004). Unsolved Problems in Number Theory, Springer-Verlag, Third Edition.
  6. Hargreaves, K. & Siksek, S. (2010). Cycles and fixed points of happy functions, Journal of Combinatorics and Number Theory, 3, 217–229.
  7. Pan, H. (2008). On consecutive happy numbers, Journal of Number Theory, 128, 1646–1654.
  8. Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, https://oeis.org
  9. Styer, R. (2010). Smallest examples of strings of consecutive happy numbers, Journal of Integer Sequences, 13, Article 10.6.3.
  10. Swart, B. B., Beck, K. A., Crook, S., Turner, C. E., Grundman, H. G., Mei, M. & Zack, L. (2017). Augmented generalized happy functions, The Rocky Mountain Journal of Mathematics, 47, 403–417.
  11. Zhou, X. & Cai, T. (2009). On e-power b-happy numbers, The Rocky Mountain Journal of Mathematics, 39, 2073–2081.

Related papers

Cite this paper

Noppakeaw, Passawan, Phoopha, Niphawan, & Pongsriiam, Prapanpong (2019). Composition of happy functions. Notes on Number Theory and Discrete Mathematics, 25(3), 13-20, DOI: 10.7546/nntdm.2019.25.3.13-20.

Comments are closed.