# Inequalities between the arithmetic functions φ, ψ and σ. Part 1

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
Full paper (PDF, 149 Kb)

## 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

• 11A25

### References

1. Atanassov, K. (1985). Short proof of a hypothesis of A. Mullin. Bulletin of Number Theory and Related Topics, IX (2), 9–11.
2. Atanassov, K. (1987). New integer functions, related to φ and σ functions. Bulletin of Number Theory and Related Topics, XI (1), 3–26.
3. Atanassov, K. (1991). Inequalities for φ and σ functions. I. Bulletin of Number Theory and Related Topics, XV, 12–14.
4. Atanassov, K. (2006). Note on φψ  and σ functions. Notes on Number Theory and Discrete Mathematics, 12 (4), 23–24.
5. Mitrinovic, D., & Sándor, J. (1996). Handbook of Number Theory, Kluwer Academic Publishers.
6. Nagell, T. (1950). Introduction to Number Theory, John Wiley & Sons, New York.
7. Sándor, J. (2019). Theory of Means and Their Inequalities. Available online: http://www.math.ubbcluj.ro/˜jsandor/lapok/Sándor-Jozsef-Theory%20of%20Means%20and%20Their%20Inequalities.pdf

## Cite this paper

Atanassov, 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.