\[n\]
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.