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On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions \(\sigma\) and \(\phi\). (English) Zbl 0965.11004

Denote by \(\varphi(n)\) and \(\sigma(n)\) Euler’s function and the sum of divisors of \(n\), respectively. A. Mąkowski and A. Schinzel [Colloq. Math. 13, 95-99 (1964; Zbl 0124.02702)] asked, if \[ \frac {\sigma(\varphi(n))} {n}\geq \frac {1}{2} \tag{\(*\)} \] holds for all positive integers \(n\). The authors show that the lower density of the set of positive integers \(n\) satisfying the inequality \((*)\) is greater than 0.74.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11N60 Distribution functions associated with additive and positive multiplicative functions

Citations:

Zbl 0124.02702
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Online Encyclopedia of Integer Sequences:

a(n) = sigma(phi(n)).
a(n) = 2*sigma(phi(n)) - n.