Amicable Numbers Calculator

This amicable numbers calculator checks whether a whole number belongs to an amicable pair, one of the oldest curiosities in number theory. Two different numbers are amicable when each one equals the sum of the other’s proper divisors, meaning every positive divisor of a number except the number itself. Enter any whole number from 2 to 1,000,000,000 and the calculator works out its divisor sum, called its partner, then checks whether that partner’s own divisor sum leads straight back to your original number. If it does, you have an amicable pair; if the partner differs from your number but does not return the favour, the number is simply not amicable, and if the partner equals the number itself it is a perfect number instead, a related but different idea. The results panel shows your number’s partner, meaning its divisor sum, alongside a clear yes-or-no verdict on whether the pair is amicable. The best-known example, recognised since antiquity, is 220 and 284: the proper divisors of each add up exactly to the other. Enter 220 to see this confirmed instantly. The calculator uses an exact integer algorithm rather than an approximation, so results stay precise even near the upper end of the range, and very large partners are displayed with grouped digits for easy reading. It is a handy tool for students, programmers and anyone exploring recreational maths and number theory.

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Whole number (2 to 1,000,000,000)

Partner (divisor sum)

Amicable?

The formula

Let s(n) be the sum of the proper divisors of n. Numbers n and m form an amicable pair when s(n) = m and s(m) = n, with n not equal to m. If s(n) = n the number is perfect, not amicable.

Worked example

The proper divisors of 220 add to 284, and the proper divisors of 284 add to 220, so 220 and 284 are amicable. Enter 220 to confirm.

Frequently asked questions

What are amicable numbers?

A pair of different numbers where each equals the sum of the other’s proper divisors, like 220 and 284.

How is this different from a perfect number?

A perfect number equals the sum of its own proper divisors. Amicable numbers point at each other instead of themselves.

What is the smallest amicable pair?

220 and 284, the pair known since ancient Greece.

Who this calculator is for

This calculator is for students, programmers and anyone exploring number theory.

What this calculator assumes

You enter a whole number within the stated range.
The exact integer algorithm is used.
Very large results are shown grouped for readability.

Formula and sources

Related calculators

Prime Factorization Calculator: break a number into prime factors.
LCM Calculator: lowest common multiple.
GCF Calculator: greatest common factor.

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