Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n) on a set of asymptotic density 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm92-1-10,
author = {Florian Luca and Carl Pomerance},
title = {On some problems of Makowski-Schinzel and Erdos concerning the arithmetical functions ph and s},
journal = {Colloquium Mathematicae},
volume = {91},
year = {2002},
pages = {111-130},
zbl = {1027.11007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm92-1-10}
}
Florian Luca; Carl Pomerance. On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ. Colloquium Mathematicae, Tome 91 (2002) pp. 111-130. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm92-1-10/