The sequence of middle divisors is unbounded

TL;DR

This paper proves that the sequence of middle divisors for natural numbers is unbounded by explicitly constructing sequences with arbitrarily many divisors in a specific range.

Contribution

It introduces a new concept of middle divisors and demonstrates their unbounded nature through explicit sequence construction.

Findings

The sequence of middle divisors is unbounded.

Explicit sequences with arbitrarily many middle divisors are constructed.

The range of interest is between rac{\sqrt{n/2}}{\sqrt{2n}}.

Abstract

The sequence of middle divisors is shown to be unbounded. For a given number $n$, $a_{n,0}$ is the number of divisors of $n$ in between $\sqrt{n/2}$ and $\sqrt{2n}$. We explicitly construct a sequence of numbers $n(i)$ and a list of divisors in the interesting range, so that the length of the list goes to infinity as $i$ increases.