On the small prime factors of a non-deficient number

TL;DR

This paper investigates the prime factorization properties of non-deficient numbers, establishing new bounds on their prime factors and analyzing related divisor sum sequences.

Contribution

It introduces new lower bounds on the number of prime factors of odd non-deficient numbers based on their smallest prime factors and refines bounds for odd perfect numbers.

Findings

New lower bounds for prime factors of odd non-deficient numbers

Tighter bounds for odd perfect numbers

Analysis of divisor sum sequences like σ(n!+1) and σ(2^n+1)

Abstract

Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. We also obtain tighter bounds for odd perfect numbers. We also discuss the behavior of $\sigma(n!+1)$, $\sigma(2^n+1)$, and related sequences.