Vol. 10, No. 3, 2021

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On a problem of De Koninck

Tomohiro Yamada

Vol. 10 (2021), No. 3, 249–260
Abstract

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 n1,1782 and σ(n) = (γ(n))2, then there exist odd (not necessarily distinct) primes p,p and (not necessarily odd) distinct primes qi (i = 1,2,,k) such that p,pn, qi2n (i = 1,2,,k), with k 3, and q1|σ(p2), qi+1|σ(qi2) (1 i k 1), p|σ(qk2).

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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