On some properties of $\sigma(N)$

TL;DR

This paper investigates the asymptotic bounds and properties of the greatest common divisor of an integer N and its sum-of-divisors function σ(N), revealing infinite instances with notably large gcd values.

Contribution

It provides new asymptotic bounds and proves the existence of infinitely many integers with large gcd of N and σ(N).

Findings

Established asymptotic upper and lower bounds for gcd(N, σ(N)).

Proved the infinitude of integers with large gcd of N and σ(N).

Enhanced understanding of the relationship between N and its sum-of-divisors.

Abstract

We show asymptotic upper and lower bounds for the greatest common divisor of N and $\sigma(N)$. We also show that there are infinitely many integers N with fairly large g.c.d. of N and $\sigma(N)$.