Amicable Numbers Checker
Check whether two numbers form an amicable pair — where each equals the sum of the other's proper divisors.
Shows all divisors and the classic known pairs.
Amicable Numbers
Two numbers a and b form an amicable pair when the sum of the proper divisors of each equals the other.
Definition
s(a) = b and s(b) = a
Where s(n) is the sum of all proper divisors of n (divisors less than n itself).
The Famous First Pair: 220 and 284
Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 → sum = 284 Proper divisors of 284: 1, 2, 4, 71, 142 → sum = 220
This pair was known to ancient Greek mathematicians and mentioned by Pythagoras.
Known Amicable Pairs (small examples)
| Pair | Sum of divisors |
|---|---|
| 220 and 284 | s(220)=284, s(284)=220 |
| 1,184 and 1,210 | Discovered in 1867 |
| 2,620 and 2,924 | Classic pair |
| 5,020 and 5,564 | Classic pair |
| 6,232 and 6,368 | Classic pair |
Perfect Numbers — A Related Concept
A perfect number satisfies s(n) = n — it is “amicable with itself.” The first perfect number is 6: 1 + 2 + 3 = 6.
History
The Pythagorean school considered 220 and 284 to represent friendship. Islamic mathematician Thabit ibn Qurra described a formula for generating amicable pairs in the 9th century. As of 2024, over 1.2 billion amicable pairs are known, but it remains unproven whether infinitely many exist.
How to Use This Calculator
Enter two numbers to check if they form an amicable pair. If you enter only the first number, the calculator finds s(a) and checks if s(s(a)) = a.