Amicable numbers come in pairs where the proper divisors of each sum to the other, as with two hundred twenty and two hundred eighty-four.

The longer version

Add the proper divisors of two hundred twenty, that is one, two, four, five, ten, eleven, twenty, twenty-two, forty-four, fifty-five, and one hundred ten, and the total is two hundred eighty-four. Reverse the process on two hundred eighty-four, whose divisors are one, two, four, seventy-one, and one hundred forty-two, and you land back on two hundred twenty.

The pair was known in antiquity and treated as a symbol of friendship, which is how it acquired its name. For centuries no other pair was found in Europe, though later searchers turned up many more, and thousands are catalogued today. Sociable chains take the idea further, cycling through more than two numbers before returning.

Where this entry sits

This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.