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

A. Grytczuk; F. Luca; M. Wójtowicz

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 1, page 31-36
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Grytczuk, A., Luca, F., and Wójtowicz, M.. "On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ." Colloquium Mathematicae 86.1 (2000): 31-36. <http://eudml.org/doc/210839>.

@article{Grytczuk2000,
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.},
author = {Grytczuk, A., Luca, F., Wójtowicz, M.},
journal = {Colloquium Mathematicae},
keywords = {composition of arithmetic functions; Euler's function; sum of divisors function; lower density},
language = {eng},
number = {1},
pages = {31-36},
title = {On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ},
url = {http://eudml.org/doc/210839},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Grytczuk, A.
AU - Luca, F.
AU - Wójtowicz, M.
TI - On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 1
SP - 31
EP - 36
AB - 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.
LA - eng
KW - composition of arithmetic functions; Euler's function; sum of divisors function; lower density
UR - http://eudml.org/doc/210839
ER -

References

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  1. [1] U. Balakrishnan, Some remark on σ(ϕ(n)), Fibonacci Quart. 32 (1994), 293-296. 
  2. [2] G. L. Cohen, On a conjecture of Mąkowski and Schinzel, Colloq. Math. 74 (1997), 1-8. 
  3. [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. [4] R. K. Guy, Unsolved Problems in Number Theory, Springer, 1994. 
  5. [5] A. Mąkowski and A. Schinzel, On the functions ϕ(n) and σ(n), Colloq. Math. 13 (1964-1965), 95-99. 
  6. [6] D. S. Mitrinović, J. Sándor and B. Crstici, Handbook of Number Theory, Kluwer, 1996. 
  7. [7] C. Pomerance, On the composition of the arithmetic functions σ and ϕ, Colloq. Math. 58 (1989), 11-15. 

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