On the small prime factors of a non-deficient number

Abstract

Let $\sigma (n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma (n)\ge 2n$. 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.

arXiv comment19 pages. Accepted to Integers