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ArticleOriginal scientific text

Title

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

Authors 1, 2, 1

Affiliations

  1. Institute of Mathematics, T. Kotarbiński Pedagogical University, 65-069 Zielona Góra, Poland
  2. Fachbereich Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

Abstract

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.

Bibliography

  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.
Colloquium Mathematicum
2000/86/1
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Pages:
31-36
Main language of publication
English
Received
1999-04-13
Accepted
1999-07-02
Published
2000
Exact and natural sciences