What is an amicable pair?

Two distinct positive integers a and b are amicable if the sum of proper divisors of a equals b, and the sum of proper divisors of b equals a. The smallest known pair is (220, 284): the proper divisors of 220 sum to 284, and the proper divisors of 284 sum to 220.

What are proper divisors?

The proper divisors of a number n are all positive divisors of n except n itself. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16.

Can a number be amicable with itself?

No — amicable pairs require two distinct numbers. A number where s(n) = n is called a perfect number (e.g., 6 and 28), not an amicable number.

How many amicable pairs are known?

Over 1.2 billion amicable pairs are known. The smallest is (220, 284), discovered in antiquity. This tool can find all pairs up to about 1,000,000 in a few seconds.

Why does the finder slow down for large limits?

Finding amicable pairs requires computing the proper divisor sum for every integer up to the limit. The algorithm runs in O(n log n) time using a sieve approach, so very large limits (above 2 million) take noticeably longer.

Amicable Number Checker

Name: Amicable Number Checker
Availability: InStock
Author: Nham Vu

Enter two numbers to check if they form an amicable pair, or find all amicable pairs up to a limit.

Check a Pair

Enter two numbers to see if they form an amicable pair.

Number A

Number B

Try a known pair

Find All Pairs up to a Limit

Lists every amicable pair where both numbers are at or below your limit.

Upper limit (max 1,000,000)

#	Number A	Number B	s(A) = B	s(B) = A

Summary

Enter two numbers to check if they form an amicable pair, or find all amicable pairs up to a limit.

How it works

Enter two numbers in the pair checker, or enter a single limit to find all pairs up to that value.
The tool computes the sum of proper divisors for each number (all divisors excluding the number itself).
Two numbers form an amicable pair if s(a) = b and s(b) = a, where s(n) is the proper divisor sum of n.
The result shows whether the pair is amicable, along with each number's proper divisors and their sums.
The finder scans every number up to your limit and reports all amicable pairs it discovers.

Use cases

Verify a specific pair of numbers for the amicability property.
Generate a reference list of amicable pairs for number theory study.
Explore the density and distribution of amicable pairs in a range.
Check homework or textbook problems about amicable numbers.
Demonstrate the concept of proper divisors to students.
Find all known small amicable pairs quickly without manual computation.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu