Note on the natural density of r-free numbers

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

1. Brown, R. (2021). The natural density of some sets of square-free numbers. Integers, A81.1-9.
2. Hardy, G. H., & Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press, Oxford.
3. Jameson, G. J. O. (2010). Even and odd square-free numbers. The Mathematical Gazette, 94, 123–127.
4. Jameson, G. J. O. (2021). Revisiting even and odd square-free numbers. The Mathematical Gazette, 105, 299–300.
5. Puttasontiphot, T., & Srichan, T. (2021). Odd/even cube-full numbers. Notes on Number Theory and Discrete Mathematics, 27(1), 27–31.
6. Scott, J. A. (2008). Square-free integers once again. The Mathematical Gazette, 92, 70–71.
7. Srichan, T. (2020). The odd/even dichotomy for the set of square-full numbers. Applied Mathematics E-Notes, 20, 528–531.
8. 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