**Sunanta Srisopha, Teerapat Srichan and Sukrawan Mavecha**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 3, Pages 435–440

DOI: 10.7546/nntdm.2022.28.3.435-440

**Full paper (PDF, 230 Kb)**

## Details

### Authors and affiliations

**Sunanta Srisopha**

*Department of Mathematics, Faculty of Science,
King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand*

**Teerapat Srichan**

*Department of Mathematics, Faculty of Science,
Kasetsart University, Bangkok 10900, Thailand*

**Sukrawan Mavecha**

*Department of Mathematics, Faculty of Science,
King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand*

### Abstract

Let *P* be a finite set of prime numbers. By using an elementary method, the proportion of all *r*-free numbers which are divisible by at least one element in *P* is studied.

### Keywords

- Natural density
*r*-free number

### 2020 Mathematics Subject Classification

- 11A41
- 11B25

### References

- Brown, R. (2021). The natural density of some sets of square-free numbers.
*Integers*, A81.1-9. - Hardy, G. H., & Wright, E. M. (1979).
*An Introduction to the Theory of Numbers*(5th ed.). Oxford University Press, Oxford. - Jameson, G. J. O. (2010). Even and odd square-free numbers.
*The Mathematical Gazette*, 94, 123–127. - Jameson, G. J. O. (2021). Revisiting even and odd square-free numbers.
*The Mathematical Gazette*, 105, 299–300. - Puttasontiphot, T., & Srichan, T. (2021). Odd/even cube-full numbers.
*Notes on Number Theory and Discrete Mathematics*, 27(1), 27–31. - Scott, J. A. (2008). Square-free integers once again.
*The Mathematical Gazette*, 92, 70–71. - Srichan, T. (2020). The odd/even dichotomy for the set of square-full numbers.
*Applied Mathematics E-Notes*, 20, 528–531. - Srisopha, S., Srichan, T., & Mavecha, S. Odd/even
*r*-free numbers.*Applied Mathematics E-Notes*(to appear).

### Manuscript history

- Received: 9 March 2022
- Revised: 24 June 2022
- Accepted: 7 July 2022
- Online First: 20 July 2022

## Related papers

## Cite this paper

Srisopha, S., Srichan, T., & Mavecha, S. (2022). Note on the natural density of *r*-free numbers. *Notes on Number Theory and Discrete Mathematics*, 28(3), 435-440, DOI: 10.7546/nntdm.2022.28.3.435-440.