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Existence of a number with a fixed number of divisors among 1,2,...,n

Source: Miklos Schweitzer 2023, Problem 6

November 7, 2023

number theory

Problem Statement

Prove that for all sufficiently large positive integers

n

and a positive integer

k \leq n

, there exists a positive integer

m

having exactly

k

divisors in the set

\{1,2, \ldots, n\}