Aleksander Grytczuk
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 1, Pages 14–18
Full paper (PDF, 152 Kb)
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* = {p ∈ P; p = 2αk + 1; α ≥ 1; k > 1; (k,2) = 1}.
Moreover, let
where (mj, mk) = 1; for all j ≠ k, 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
AMS Classification
- 11A25
- 11N37
References
- Balakrishnan, U. Some remark on σ(φ(n)), Fibonacci Quarterly, Vol. 32, 1994, 293–296.
- Cohen, G.L. On a conjecture of Mąkowski and Schinzel, Colloq. Math., Vol. 74, 1997, 1–8.
- Filaseta, M., S. W. Graham, C. Nicol, On the composition of σ(n) and φ(n), Abstracts AMS, Vol. 13, 1992, No.4, 137.
- Ford, K. An explicit sieve bound and small values of σ(φ(n)), Period. Math. Hung, Vol. 43, 2001, 15–29.
- Grytczuk, A., F. Luca, M. Wójtowicz, On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and φ, Colloq. Math., Vol. 86, 2000, 31–36.
- Guy, R. K. Unsolved Problems in Number Theory, Springer-Verlag, 1994
- Luca, F., C. Pomerance, On some problems of Mąkowski and Erdos concerning the arithmetic functions φ and σ, Colloq. Math., Vol. 92, 2002, 111–130.
- Mąkowski. A., A. Schinzel, On the functions φ (n) and σ(n), Colloq. Math., Vol. 13, 1964/65, 95–99.
- Redmond, D. Number Theory, Marcel Dekker, Inc., New York, 1996.
- Sándor, J. On the composition of some arithmetic functions, Studia Babes-Bolyai University, Mathematica, Vol. 34, 1989, 7–14.
- Sándor, J. Remarks on the functions σ(n) and φ(n), Babes-Bolyai Univ. Seminar on Math. Analysis, Vol. 7, 1989, 7–12.
- Sándor, J. A note on the functions σ(n) and φ(n), Studia Babes-Bolyai University, Mathematica, Vol. 35, 1990, 3–6.
- Sándor, J., B. Crstici, Handbook of Number Theory, II, Kluwer Academic Publishers, Dordrecht / Boston / London, 2004.
- Sándor, J. On the composition of some arithmetic functions, II. J. Inequal. Pure and Appl. Math., Vol. 6, 2005, No. 3, Art. 73.
Related papers
Cite this paper
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.