Let φ(·) and σ(·) denote the Euler function and the sum of divisors function, respectively. We give a lower bound for the number of m ≤ x for which the equation m = σ(n) - n has no solution. We also show that the set of positive integers m not of the form (p-1)/2 - φ(p-1) for some prime number p has a positive lower asymptotic density.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm103-1-4,
author = {William D. Banks and Florian Luca},
title = {Nonaliquots and Robbins numbers},
journal = {Colloquium Mathematicae},
volume = {103},
year = {2005},
pages = {27-32},
zbl = {1089.11007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm103-1-4}
}
William D. Banks; Florian Luca. Nonaliquots and Robbins numbers. Colloquium Mathematicae, Tome 103 (2005) pp. 27-32. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm103-1-4/