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Inequality with divisors

Source: Russian TST 2019, Day 7 P1 (Groups A & B)

March 27, 2023
number theoryDivisors

Problem Statement

A positive integer nn{} is called discontinuous if for all its natural divisors 1=d0<d1<⋯<dk1 = d_0 < d_1 <\cdots<d_k, written out in ascending order, there exists 1⩽i⩽k1 \leqslant i \leqslant k such that di>di−1+⋯+d1+d0+1d_i > d_{i-1}+\cdots+d_1+d_0+1. Prove that there are infinitely many positive integers nn{} such that n,n+1,…,n+2019n,n+1,\ldots,n+2019 are all discontinuous.
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