A Family of Iterated Maps on Natural Numbers

Abstract

In this paper, we introduce and study the iterates of the following family of functions ${\phi }_k$ defined on natural numbers which exhibits nice properties.

${\phi }_k(x)=\{\begin{array}{cc}x+k,\hfill & \text{ if }x\text{ is prime};\hfill \\ \text{largest prime divisor of }x,\hfill & \text{ if }x\text{ is composite};\hfill \end{array}$

In particular, we study the periodic behavior of the trajectories of these iterated functions. In some cases, we provide proofs of these properties and in some other cases we pose some open problems based on numerical evidences supported by heuristic arguments.

KEYWORDS:

• Arithmetic functions
• special sequences

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 11B83
• 11B37
• 11A25

Acknowledgments

The author is also thankful to Hiranya Kishore Dey and Anubrato Bhattacharya for various fruitful discussions, and the weekly departmental seminar series QED of Presidency University for providing a stimulus to think about this problem.

Additional information

Funding

The author acknowledge the funding of DST-FIST Sanction no. SR/FSST/MS–I/2019/41 and DST-SERB-MATRICS Sanction no. MTR/2022/00020, Govt. of India.