On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ

• Autorzy: A. Grytczuk , F. Luca , M. Wójtowicz
• Języki publikacji: EN

Abstrakty

EN

For any positive integer n let ϕ(n) and σ(n) be the Euler function of n and the sum of divisors of n, respectively. In [5], Mąkowski and Schinzel conjectured that the inequality σ(ϕ(n)) ≥ n/2 holds for all positive integers n. We show that the lower density of the set of positive integers satisfying the above inequality is at least 0.74.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2000
• Tom: 86
• Numer: 1
• Strony: 31-36

Daty

wydano

2000

otrzymano

1999-04-13

poprawiono

1999-07-02

Twórcy

autor

A. Grytczuk

• Institute of Mathematics, T. Kotarbiński Pedagogical University, 65-069 Zielona Góra, Poland

autor

F. Luca

• Fachbereich Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

autor

M. Wójtowicz

• Institute of Mathematics, T. Kotarbiński Pedagogical University, 65-069 Zielona Góra, Poland

Bibliografia

• [1] U. Balakrishnan, Some remark on σ(ϕ(n)), Fibonacci Quart. 32 (1994), 293-296.
• [2] G. L. Cohen, On a conjecture of Mąkowski and Schinzel, Colloq. Math. 74 (1997), 1-8.
• [3] M. Filaseta, S. W. Graham and C. Nicol, On the composition of σ(n) and ϕ(n), Abstracts Amer. Math. Soc. 13 (1992), no. 4, p. 137.
• [4] R. K. Guy, Unsolved Problems in Number Theory, Springer, 1994.
• [5] A. Mąkowski and A. Schinzel, On the functions ϕ(n) and σ(n), Colloq. Math. 13 (1964-1965), 95-99.
• [6] D. S. Mitrinović, J. Sándor and B. Crstici, Handbook of Number Theory, Kluwer, 1996.
• [7] C. Pomerance, On the composition of the arithmetic functions σ and ϕ, Colloq. Math. 58 (1989), 11-15.