MathDB
existence of a sequence

Source: 2018 Balkan MO Shortlist N1

May 22, 2019
number theoryBalkan MO Shortlist

Problem Statement

For positive integers mm and nn, let d(m,n)d(m, n) be the number of distinct primes that divide both mm and nn. For instance, d(60,126)=d(22⋅3⋅5,2⋅32⋅7)=2.d(60, 126) = d(2^2 \cdot 3 \cdot 5, 2 \cdot 3^2 \cdot 7) = 2. Does there exist a sequence (an)(a_n) of positive integers such that: [*] a1≥20182018;a_1 \geq 2018^{2018}; [*] am≤ana_m \leq a_n whenever m≤nm \leq n; [*] d(m,n)=d(am,an)d(m, n) = d(a_m, a_n) for all positive integers m≠nm\neq n?
(Dominic Yeo, United Kingdom)
Back to ProblemsView on AoPS