# On the composition of the functions σ and φ on the set Zs+(P*)

Aleksander Grytczuk
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 19, 2013, Number 1, Pages 14—18

## Details

### Authors and affiliations

Aleksander Grytczuk
Faculty of Mathematics, Computer Science and Econometrics
University of Zielona Góra
4a Prof. Szafrana Str., 65-516 Zielona Góra, Poland

### Abstract

In 1964, A. Mąkowski and A. Schinzel ([8], Cf.[6]) conjectured that for all positive integers m, we have
(*)
where σ denote the sum of divisors function and φ is the Euler’s totient function.
Let P be the set of all odd primes and
P* = {pP; p = 2αk + 1; α ≥ 1; k > 1; (k,2) = 1}.
Moreover, let

where (mj, mk) = 1; for all jk, j, k = 1, 2, …, r. In this paper we prove that if n
then we have . From this and Sándor’s result it follows that (*) is true for all positive integers m ≥ 1 such that the squarefree part of m.

### Keywords

• Arithmetical functions
• Mąkowski—Schinzel conjecture

• 11A25
• 11N37

### References

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## Cite this paper

APA

Grytczuk, A. (2013). On the composition of the functions σ and φ on the set Zs+(P*), Notes on Number Theory and Discrete Mathematics, 19(1), 14-18.

Chicago

Grytczuk, Aleksander. “On the Composition of the Functions σ and φ on the Set Zs+(P*).” Notes on Number Theory and Discrete Mathematics 19, no. 1 (2013): 14-18.

MLA

Grytczuk, Aleksander. “On the Composition of the Functions σ and φ on the Set Zs+(P*).” Notes on Number Theory and Discrete Mathematics 19.1 (2013): 14-18. Print.