Grytczuk, A.; Luca, F.; Wójtowicz, M. On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions \(\sigma\) and \(\phi\). (English) Zbl 0965.11004 Colloq. Math. 86, No. 1, 31-36 (2000). 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. Reviewer: Thomas Maxsein (Clausthal) Cited in 1 Review MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11N60 Distribution functions associated with additive and positive multiplicative functions Keywords:composition of arithmetic functions; Euler’s function; sum of divisors function; lower density Citations:Zbl 0124.02702 PDFBibTeX XMLCite \textit{A. Grytczuk} et al., Colloq. Math. 86, No. 1, 31--36 (2000; Zbl 0965.11004) Full Text: DOI EuDML Online Encyclopedia of Integer Sequences: a(n) = sigma(phi(n)). a(n) = 2*sigma(phi(n)) - n.