Highly Composite Numbers

Abstract

The main result of this thesis is to show that there are only finitely many integers $n$ such that both $n$ and $d(n)$ are highly composite numbers at the same time, where $d(n)$ is the divisor function. Bertrand's postulate [4] is used many times throughout the thesis and allows us to write a proof that is as simple (and as short) as possible. This thesis is meant to solve the open problem from the ``On-Line Encyclopedia of Integer Sequences" (OEIS): A189394 [3]. The main idea for solving the problem comes from the comment in A189394; $n$ will contain many primes with exponent 1 when $n$ is a large highly composite number. This implies that $d(n)$ contains a lot of factors of 2. We then estimate the factor $2^{\beta_1}$ in $d(n)$ in terms of the largest prime in $d(n)$ from above and from below to give us a contradiction when $n$ is large enough. We end by finding a list of all highly composite $n$ such that $d(n)$ is also highly composite.