# 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

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topGrytczuk, 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

top- [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.

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