Amicable Numbers Checker

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)

The 1,184 and 1,210 pair is the one worth pausing on. It is the second-smallest pair there is, small enough that Euler and Fermat and everyone else had walked straight past it for centuries. Paganini was sixteen when he spotted it, apparently by simple trial. Nobody has ever explained how everyone else missed it.

Perfect numbers, a related idea

A perfect number satisfies s(n) = n, so it is amicable with itself. The first is 6, since 1 + 2 + 3 = 6. The next three are 28, 496 and 8,128. This calculator will not report a perfect number as an amicable pair, because a pair needs two different numbers. It says so explicitly when you enter one twice.

A number is one of three things

Compare s(n) against n and every positive integer falls into one bucket. If s(n) is less than n the number is deficient, which covers all primes (s(p) = 1) and most numbers you will try. If s(n) is greater it is abundant, and 12 is the smallest of those (1+2+3+4+6 = 16). Equality makes it perfect, and those are rare enough that only 52 were known as of 2026. The calculator labels both of your numbers this way, because the same divisor sum answers all of it for free.

History

The Pythagorean school treated 220 and 284 as a symbol of friendship, and the pair turns up in medieval talismans given between lovers. Thabit ibn Qurra, working in Baghdad in the 9th century, described a formula that generates amicable pairs from three primes of a particular form. Over a billion pairs are known today, found by exhaustive computer search, yet nobody has proved whether the supply is infinite.

How to use this calculator

Enter two numbers to check whether they form an amicable pair. Enter only the first and it computes s(a), then checks whether s(s(a)) brings you back, which is how you hunt for a partner rather than verify one.