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
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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

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Cite this paper

APA

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.

Chicago

Noppakeaw, Passawan, Phoopha, Niphawan, & Pongsriiam, Prapanpong. “Composition of happy functions.” Notes on Number Theory and Discrete Mathematics 25, no. 3 (2019): 13-20, doi: 10.7546/nntdm.2019.25.3.13-20.

MLA

Noppakeaw, Passawan, Phoopha, Niphawan, & Pongsriiam, Prapanpong. “Composition of happy functions.” Notes on Number Theory and Discrete Mathematics 25.3 (2019): 13-20. Print, doi: 10.7546/nntdm.2019.25.3.13-20.

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