(a) Show that $n$ is an abundant number $\iff 𝜎 (n) > 2n$.
(b) Show that $n$ is a deficient number $\iff 𝜎 (n) < 2n$.
(c) Show that $n$ is a perfect number $\iff 𝜎 (n) = 2n$.
Characterise the numbers in question 2 into perfect, abundant, or deficient numbers.
(a) An abundant number $n$ is one where the sum of its proper factors $p(n)$ is greater than $n$.
Noting the definition $\sigma(n) = p(n) + n$, we have
$$ \text{abundant }n \iff p(n) > n \iff \sigma(n) > n + n \iff \sigma(n) > 2n $$
(b) A deficient number $n$ is one where the sum of its proper factors $p(n)$ is less than $n$.
Noting the definition $\sigma(n) = p(n) + n$, we have
$$ \text{deficient }n \iff p(n) < n \iff \sigma(n) < n + n \iff \sigma(n) < 2n $$
(c) A perfect number $n$ is one where the sum of its proper factors $p(n)$ equals $n$.
Noting the definition $\sigma(n) = p(n) + n$, we have
$$ \text{perfect }n \iff p(n) = n \iff \sigma(n) = n + n \iff \sigma(n) = 2n $$
We characterise the numbers from exercise 2 as follows:
- 15 is a deficient number because $\sigma(15) = 24 < 30$
- 77 is a deficient number because $\sigma(77)=96 < 154$
- 171 is a deficient number because $\sigma(171) = 260 < 342$
- 200 is an abundant number because $\sigma(200)=465 > 400$