Krassimir T. Atanassov and József Sándor

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 1, Pages 50—53

DOI: 10.7546/nntdm.2019.25.1.50-53

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## Details

### Authors and affiliations

Krassimir T. Atanassov

*Department of Bioinformatics and Mathematical Modelling
Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences,
Acad. G. Bonchev Str., Bl. 105, Sofia-1113, Bulgaria*

*Intelligent Systems Laboratory, Prof. Asen Zlatarov University
Bourgas-8010, Bulgaria*

József Sándor

*Department of Mathematics, Babeș–Bolyai University
Str. Kogalniceanu 1, 400084 Cluj-Napoca, Romania
*

### Abstract

In this article we determine the minimal set for some sets of natural numbers. The concept of minimal sets (in the context of natural numbers) appeared first in an article of Shallit, who determined, among others, the minimal set of the primes. By now, there are several articles about minimal sets. In this article we will expand results of Baoulina, Kreh and Steuding, who determined the minimal set of the sets *φ*(ℕ) and *φ*(ℕ) + 3. To this end, we will determine the minimal set of the sets *φ*(ℕ) + *a* for 1 ≤ *a* ≤ 5.

### Keywords

- Arithmetic function
- Inequality

### 2010 Mathematics Subject Classification

- 11A25

### References

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*φ*and*σ*functions. Bulletin of Number Theory and Related Topics, XI (1), 3–26. - Atanassov, K. (1991). Inequalities for
*φ*and*σ*functions. I. Bulletin of Number Theory and Related Topics, XV, 12–14. - Atanassov, K. (2006). Note on
*φ*,*ψ*and*σ*functions. Notes on Number Theory and Discrete Mathematics, 12 (4), 23–24. - Mitrinovic, D., & Sándor, J. (1996). Handbook of Number Theory, Kluwer Academic Publishers.
- Nagell, T. (1950). Introduction to Number Theory, John Wiley & Sons, New York.
- Sándor, J. (2019). Theory of Means and Their Inequalities. Available online: http://www.math.ubbcluj.ro/˜jsandor/lapok/Sandor-Jozsef-Theory%20of%20Means%20and%20Their%20Inequalities.pdf

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## Cite this paper

APAAtanassov, K. & Sándor, J. (2019). Inequalities between the arithmetic functions *φ*, *ψ* and *σ*. Part 1. Notes on Number Theory and Discrete Mathematics, 25(1), 50-53, doi: 10.7546/nntdm.2019.25.1.50-53.

Atanassov, Krassimir and József Sándor. “Inequalities between the Arithmetic Functions *φ*, *ψ* and *σ*. Part 1.” Notes on Number Theory and Discrete Mathematics 25, no. 1 (2019): 50-53, doi: 10.7546/nntdm.2019.25.1.50-53.

Atanassov, Krassimir and József Sándor. “Inequalities between the Arithmetic Functions *φ*, *ψ* and *σ*. Part 1.” Notes on Number Theory and Discrete Mathematics 25.1 (2019): 50-53. Print, doi: 10.7546/nntdm.2019.25.1.50-53.