  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.
