On a problem of De Koninck

Let $σ (n)$ denote the sum of divisors of $n$ and $γ (n)$ denote the product of distinct prime divisors of $n$ . We shall show that, if $n \neq 1, 1782$ and $σ (n) = {(γ (n))}^{2}$ , then there exist odd (not necessarily distinct) primes $p, p^{'}$ and (not necessarily odd) distinct primes $q_{i}$ ( $i = 1, 2, \dots, k$ ) such that $p, p^{'} ∥ n$ , $q_{i}^{2} ∥ n$ ( $i = 1, 2, \dots, k$ ), with $k \leq 3$ , and $q_{1} | σ (p^{2})$ , $q_{i + 1} | σ (q_{i}^{2})$ ( $1 \leq i \leq k - 1$ ), $p^{'} | σ (q_{k}^{2})$ .

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Keywords

sum of divisors, square-free core, radical of an integer, De Koninck's conjecture, directed multigraphs, directed acyclic multigraphs

Mathematical Subject Classification

Primary: 11A25

Secondary: 05C20, 11A05, 11A41

Milestones

Received: 31 March 2021

Revised: 31 July 2021

Accepted: 25 August 2021

Published: 13 September 2021

Authors

Tomohiro Yamada
Center for Japanese Language and Culture Osaka University Osaka Japan

Vol. 10, No. 3, 2021

Tomohiro Yamada

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors