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

**Download 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 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

- El-Sedy, E. & Siksek, S. (2000). On happy numbers, The Rocky Mountain Journal of Mathematics, 30, 565–570.
- Gilmer, J. (2011). On the density of happy numbers, posted at: http://arxiv.org/pdf/1110.3836v3.pdf
- Grundman, H. G. & Teeple, E. A. (2001). Generalized happy numbers, The Fibonacci Quarterly, 39, 462–466.
- Grundman, H. G. & Teeple, E. A. (2008). Iterated sums of fifth powers of digits, The Rocky Mountain Journal of Mathematics, 38, 1139–1146.
- Guy, R. K. (2004). Unsolved Problems in Number Theory, Springer-Verlag, Third Edition.
- Hargreaves, K. & Siksek, S. (2010). Cycles and fixed points of happy functions, Journal of Combinatorics and Number Theory, 3, 217–229.
- Pan, H. (2008). On consecutive happy numbers, Journal of Number Theory, 128, 1646–1654.
- Sloane, N. J. A. The On-Line Encyclopedia of Integer Sequences, https://oeis.org
- Styer, R. (2010). Smallest examples of strings of consecutive happy numbers, Journal of Integer Sequences, 13, Article 10.6.3.
- 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.
- 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.