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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## 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 and , let be defined by

if is the expansion of in base . We call an -happy function. Let be a composition of various -happy functions. We show that, for any given , the iteration sequence either converges to a fixed point or eventually becomes a cycle. Here is the identity function mapping to for all and is the -fold composition of . 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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## Related papers

## Cite this paper

APANoppakeaw, 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.

ChicagoNoppakeaw, 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.

MLANoppakeaw, 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.