Amicable Number Checker

What Are Amicable Numbers?

Two distinct positive integers $a$ and $b$ form an amicable pair if the sum of the proper divisors of each equals the other. In mathematical notation:

where $s(n)$ is the sum of all positive divisors of $n$ excluding $n$ itself. The smallest and most famous amicable pair is $(220, 284)$:

• Proper divisors of 220: $1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284$
• Proper divisors of 284: $1 + 2 + 4 + 71 + 142 = 220$

Each number "generates" the other through its own divisors, creating a perfect two-way relationship known since the time of Pythagoras.

How to Use the Amicable Number Checker

Enter one or two positive integers into the fields above. If you enter two numbers, the tool checks whether they form an amicable pair by computing and comparing their proper divisor sums. If you enter just one number, it automatically computes its proper divisor sum to find the candidate partner, then verifies whether the pair is truly amicable.

Click any of the preset buttons to try famous amicable pairs like (220, 284) discovered by Pythagoras, (1184, 1210) found by the 16-year-old Niccolò Paganini, or (2620, 2924) discovered by Euler. You can also test numbers that are NOT amicable to see how the verification fails.

A Short History of Amicable Numbers

Amicable numbers have fascinated mathematicians for over 2,500 years. Pythagoras (c. 500 BC) knew the pair (220, 284) and called it a symbol of friendship. In the 9th century, Thabit ibn Qurra discovered the first general rule for generating amicable pairs. Fermat and Descartes independently rediscovered pairs in the 17th century, and Euler vastly expanded the list, discovering 59 new pairs.

Perhaps the most remarkable story is that of Niccolò Paganini, a 16-year-old Italian student who discovered (1184, 1210) in 1866 — the second-smallest amicable pair, which every great mathematician before him had missed. Today, collaborative computing projects have found more than 1.2 billion amicable pairs.

Thabit ibn Qurra's Rule

Around 850 AD, Thabit ibn Qurra discovered a partial formula for generating amicable pairs:

If $p$, $q$, and $r$ are all prime, then $(2^n \cdot p \cdot q, \; 2^n \cdot r)$ is an amicable pair. Setting $n = 2$ gives $p = 5$, $q = 11$, and $r = 71$ — all prime — producing the classical pair (220, 284).

Aliquot Sequences and Sociable Numbers

The aliquot sequence of a number $n$ is the sequence obtained by repeatedly applying the proper divisor sum. Amicable pairs form 2-cycles in this sequence. Longer cycles are called sociable numbers — for example, the 5-cycle starting at 12496. Perfect numbers form fixed points where $s(n) = n$, and aspiring numbers eventually reach a perfect number and terminate.

First Ten Amicable Pairs

Related Tools

If you found this amicable number checker useful, you might also enjoy our Number Divisors Calculator for finding all divisors of any number, our Prime Factorization Calculator for breaking numbers into their prime factors, or our Perfect Number Checker for finding numbers that equal the sum of their own proper divisors.