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

A. Grytczuk, F. Luca, M. Wójtowicz (2000)

Colloquium Mathematicae

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.

On the composition of the Euler function and the sum of divisors function

Jean-Marie De Koninck, Florian Luca (2007)

Colloquium Mathematicae

Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.

On the concentration of certain additive functions

Dimitris Koukoulopoulos (2014)

Acta Arithmetica

We study the concentration of the distribution of an additive function f when the sequence of prime values of f decays fast and has good spacing properties. In particular, we prove a conjecture by Erdős and Kátai on the concentration of f ( n ) = p | n ( l o g p ) - c when c > 1.

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