# On integers $n$ for which $\sigma(2n+1)\geq \sigma(2n)$

## TL;DR

This paper investigates the frequency of integers n where the sum-of-divisors function of 2n+1 exceeds or equals that of 2n, establishing that this occurs with a natural density around 5.4%.

## Contribution

It provides bounds on the natural density of integers n satisfying the inequality involving the sum-of-divisors function for 2n and 2n+1.

## Key findings

- Natural density of such n is between 0.053 and 0.055.
- The result quantifies how often the inequality holds.
- Provides bounds on the distribution of these integers.

## Abstract

We show that the natural density of positive integers $n$ for which $\sigma(2n+1)\geq \sigma(2n)$ is between $0.053$ and $0.055$.