Sums of proper divisors with missing digits

arXiv (Cornell University)

Abstract

Let $s(n)$ denote the sum of proper divisors of an integer $n$. In 1992, Erdős, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also has asymptotic density zero. In this paper we show that the EGPS conjecture holds when $\mathcal{A}$ is taken to be a set of integers with missing digits. In particular, we give a sharp upper bound for the size of this preimage set. We also provide an overview of progress towards the EGPS conjecture and survey recent work on sets of integers with missing digits.

Authors 5

1. Kübra Benli̇

   University of Alberta

   Affiliation as printed

   University of Alberta (116 St & 85 Ave, Edmonton, AB T6G 2R3 - Canada)

2. Giulia Cesana

   University of Cologne

   Affiliation as printed

   University of Cologne, Department of Mathematics and Computer Science, Division of Mathematics, Weyertal 86-90, 50931 Cologne, Germany (Germany)

3. Cécile Dartyge

   Université de Lorraine · Institut Élie Cartan de Lorraine

   Affiliation as printed

   IECL - Institut Élie Cartan de Lorraine (Université de Lorraine, Boulevard des Aiguillettes BP 70239 54506 Vandoeuvre-les-Nancy Cedex Ile du Saulcy - 57 045 Metz Cedex 01 - France)

4. Charlotte Dombrowsky Aachen

   Leiden University

   Affiliation as printed

   Universiteit Leiden, Science Mathematisch Instituut, Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, Netherlands (Netherlands)

5. Lola Thompson

   Utrecht University

   Affiliation as printed

   Universiteit Utrecht / Utrecht University [Utrecht] (Heidelberglaan 8, 3584 CS Utrecht - Netherlands)