Abundant and Deficient Numbers

An integer  
\[n\]
  is said to be abundant if  
\[\sigma (n) \gt 2n\]
  and deficient if  
\[\sigma (n) \lt 2n\]
  where  
\[\sigma (n)\]
  is the sum of the divisors of  
\[n\]
, including 1 and  
\[n\]
.
Every power of a prime number is deficient since  
\[\sigma (p^k)=1+p+...+p^k =\frac{p^{k+1}-1}{p-1} \lt 2p^k\]
.
Every product of two prime numbers  
\[n=pq\]
  is deficient since  
\[\sigma (pq)=1+p+q+pq \le 2pq\]
,
\[105^2\]
  is abundant.  
\[\begin{equation} \begin{aligned} \sigma(105^2) &= \sigma (3^2 5^2 7^2) \\ &= \sigma (3^2) \sigma (5^2) \sigma (7^2) \\ &= (1+3+3^2)(1+5+5^2)(1+7+7^2) \\ &=22971 \gt 22050=105^2 \end{aligned} \end{equation}\]
.
If  
\[m\]
  is abundant and  
\[gcd(mn)=1\]
  then  
\[\sigma(mn)= \sigma (m) \sigma (n) \lt 2m \sigma (n) \le 2m (n+1)\]
  so  
\[mn\]
  is abundant.
\[\sigma (2^{k-1}(2^k-1))=(2^k-1) \sigma (2^k-1) \gt (2^k-1)2^k=2n\]
  if  
\[2^k-1\]
  is not prime.