A positive integer \(d\) is a divisor of \(n\) if \(d\) divides \(n\) exactly:
\[
\begin{aligned}
d\mid n
\quad\Longleftrightarrow\quad
n=dq
\end{aligned}
\]
2. Sum of proper divisors
The sum of proper divisors is written \(s(n)\):
\[
\begin{aligned}
s(n)
&=
\sum_{d\in D(n)}d.
\end{aligned}
\]
In words, \(s(n)\) means: list all positive proper divisors of \(n\), then add them.
Example:
\[
\begin{aligned}
D(10)&=\{1,2,5\},\\
s(10)&=1+2+5\\
&=8.
\end{aligned}
\]
3. Definition of amicable numbers
Two distinct positive integers \(a\) and \(b\) form an amicable pair if:
\[
\begin{aligned}
s(a)&=b,\\
s(b)&=a,\\
a&\ne b.
\end{aligned}
\]
The condition \(a\ne b\) is important. It separates amicable pairs from perfect numbers.
4. The classic example: 220 and 284
First find the proper divisors of \(220\):
\[
\begin{aligned}
D(220)
&=
\{1,2,4,5,10,11,20,22,44,55,110\}.
\end{aligned}
\]
Add them:
\[
\begin{aligned}
s(220)
&=1+2+4+5+10+11+20+22+44+55+110\\
&=284.
\end{aligned}
\]
Now find the proper divisors of \(284\):
\[
\begin{aligned}
D(284)
&=
\{1,2,4,71,142\}.
\end{aligned}
\]
Add them:
\[
\begin{aligned}
s(284)
&=1+2+4+71+142\\
&=220.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
s(220)&=284,\\
s(284)&=220.
\end{aligned}
\]
Since \(220\ne284\), the pair \((220,284)\) is amicable.
5. Why perfect numbers are not amicable pairs with themselves
A perfect number satisfies
\[
\begin{aligned}
s(n)&=n.
\end{aligned}
\]
For example:
\[
\begin{aligned}
D(6)&=\{1,2,3\},\\
s(6)&=1+2+3=6.
\end{aligned}
\]
However, \((6,6)\) is not an amicable pair, because amicable numbers must be two distinct positive integers.
6. Non-amicable example
Consider \(a=10\) and \(b=16\).
\[
\begin{aligned}
D(10)&=\{1,2,5\},&
s(10)&=8,\\
D(16)&=\{1,2,4,8\},&
s(16)&=15.
\end{aligned}
\]
For the pair to be amicable, we would need \(s(10)=16\) and \(s(16)=10\). Instead:
\[
\begin{aligned}
s(10)&=8\ne16,\\
s(16)&=15\ne10.
\end{aligned}
\]
So \(10\) and \(16\) are not amicable.
7. Efficient divisor search
Divisors come in pairs. If \(d\mid n\), then \(n/d\) is also a divisor.
For example:
\[
\begin{aligned}
4\mid 220
\quad\Longrightarrow\quad
\frac{220}{4}=55
\end{aligned}
\]
so both \(4\) and \(55\) are divisors of \(220\).
This is why it is enough to test possible divisors up to \(\sqrt n\).
\[
\begin{aligned}
d\le\sqrt n.
\end{aligned}
\]
When a divisor \(d\) is found, the paired divisor \(n/d\) is found at the same time.
8. Step-by-step method
- List the proper divisors of \(a\).
- Add them to compute \(s(a)\).
- List the proper divisors of \(b\).
- Add them to compute \(s(b)\).
- Check whether \(s(a)=b\), \(s(b)=a\), and \(a\ne b\).
9. Known amicable pairs
Here are some common amicable pairs. The table uses plain text formulas so raw LaTeX cannot appear inside table cells.
| Pair |
Divisor-sum relationship |
Classification |
| 220 and 284 |
s(220) = 284 and s(284) = 220 |
Amicable |
| 1184 and 1210 |
s(1184) = 1210 and s(1210) = 1184 |
Amicable |
| 2620 and 2924 |
s(2620) = 2924 and s(2924) = 2620 |
Amicable |
| 5020 and 5564 |
s(5020) = 5564 and s(5564) = 5020 |
Amicable |
| 6232 and 6368 |
s(6232) = 6368 and s(6368) = 6232 |
Amicable |
10. Formula summary
The table below avoids LaTeX in table cells to prevent rendering problems in templates where MathJax does not process table content reliably.
11. Common mistakes
- Including the number itself in the proper divisor sum.
- Forgetting the distinct condition \(a\ne b\).
- Thinking that a perfect number paired with itself is amicable.
- Checking only \(s(a)=b\) but forgetting to check \(s(b)=a\).
- Missing paired divisors greater than \(\sqrt n\).
Key idea: an amicable pair is a two-way divisor-sum match: \(a\) sends to \(b\), and \(b\) sends back to \(a\).
