What this calculator does
An amicable pair is two different numbers where each equals the sum of the other's proper divisors. The smallest pair, 220 and 284, was known to the Pythagoreans and carried mystical significance for centuries.
They are only slightly less rare than perfect numbers. The next pair is 1184 and 1210, which was not discovered until 1866, and by a sixteen year old at that, despite mathematicians having searched for two thousand years.
The formula
The proper divisors of each number are summed. The pair is amicable when each sum equals the other number, and the two numbers are different.
| Term | Meaning |
|---|---|
| Amicable pair | Two distinct numbers each equal to the sum of the other's proper divisors. |
| Proper divisors | All divisors below the number itself. |
| Sociable numbers | The generalisation to longer cycles of three or more numbers. |
The inputs explained
| Field | What to enter |
|---|---|
| First number | The first number of the candidate pair. |
| Second number | The second number. Must differ from the first, since equal values would be testing for a perfect number. |
When to use it
Testing a candidate pair
Verifying an amicable pair requires two divisor sums.
Exploring number theory
Amicable numbers are a natural extension of perfect numbers.
Checking a historical claim
Many proposed pairs over the centuries turned out to be wrong.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Which partner makes 220 amicable?
Three candidate partners tested.
Questions
Why is 220 and 284 the famous pair?
Because it is the smallest and has been known since antiquity. The Pythagoreans attributed symbolic meaning to it, and it appears in mystical and religious writings across several cultures as a symbol of friendship.
How many amicable pairs are known?
Many millions now, found by computer search. For most of history only a handful were known: Fermat and Descartes each contributed one in the seventeenth century, and Euler found dozens more.
What is the 1184 and 1210 story?
It is the second smallest pair and was overlooked entirely until 1866, when a sixteen year old Italian named Nicolò Paganini found it. Mathematicians including Euler had searched for centuries and missed something that small.
What are sociable numbers?
The same idea extended to longer cycles. Three or more numbers where each divisor sum leads to the next and the last returns to the first. Cycles of length 4, 5, 6, 9 and 28 are known.
For a number equal to its own divisor sum, see the perfect number calculator. For divisibility rules, see the divisibility test calculator.