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n has at least k distinct prime divisors, n | 2^{\sigma(n)}-1

Source: 2015 VMEO IV Seniors 12.2 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

September 19, 2021

number theorysum of divisorsprime divisor

Problem Statement

Given a positive integer

k

. Prove that there are infinitely many positive integers

n

satisfy the following conditions at the same time: a)

n

has at least

k

distinct prime divisors b) All prime divisors other than

3

n

have the form

4t+1

, with

t

some positive integer. c)

n | 2^{\sigma(n)}-1

Here

\sigma(n)

demotes the sum of the positive integer divisors of

n