Amicable Pair — Definition, Formula & Examples

An amicable pair is two different whole numbers where the sum of the proper divisors of each number equals the other number. The most famous example is 220 and 284.

Two distinct positive integers $a$ and $b$ form an amicable pair if $s(a) = b$ and $s(b) = a$ , where $s(n)$ denotes the sum of all proper divisors of $n$ (all positive divisors of $n$ excluding $n$ itself).

Key Formula

s(a) = b \quad \text{and} \quad s(b) = a

Where:

$a, b$ = Two distinct positive integers forming the pair
$s(n)$ = The sum of all proper divisors of n

How It Works

To check whether two numbers form an amicable pair, find all proper divisors of each number and sum them. If the sum of the proper divisors of the first number equals the second, and vice versa, they are amicable. A proper divisor of

n

is any positive integer that divides

n

evenly, other than

n

itself. Note that a perfect number (like 6 or 28) is a number whose proper divisors sum to itself — you can think of it as being "amicable with itself," though by convention amicable pairs require two distinct numbers.

Worked Example

Problem: Verify that 220 and 284 form an amicable pair.

Step 1: Find all proper divisors of 220. These are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110.

s(220) = 1+2+4+5+10+11+20+22+44+55+110 = 284

Step 2: Find all proper divisors of 284. These are 1, 2, 4, 71, and 142.

s(284) = 1+2+4+71+142 = 220

Step 3: Since

s(220) = 284

and

s(284) = 220

, the condition is satisfied in both directions.

Answer: 220 and 284 are an amicable pair because the proper divisor sum of each equals the other.

Why It Matters

Amicable pairs appear in number theory courses and math competitions as a classic application of divisor functions. They were studied by Pythagoras, Euler, and many others, making them a gateway into deeper topics like aliquot sequences and the distribution of divisors. Over 1.2 billion amicable pairs are known, yet whether infinitely many exist remains an open problem.

Common Mistakes

Mistake: Including the number itself when summing its divisors.

Correction: Only sum the proper divisors — every positive divisor except the number itself. For example, the divisors of 284 are 1, 2, 4, 71, 142, and 284, but you exclude 284 from the sum.