A number that evenly divides a natural number is called a divisor. Take all of a number's divisors, drop only itself, and gather up the rest, and you get its "proper divisors." 12's divisors, for example, are 1, 2, 3, 4, 6, 12 — but its proper divisors are 1, 2, 3, 4, 6, leaving out 12. Add up all these proper divisors, and what do you get? The answer might be smaller than the original number, larger than it, or — remarkably — exactly the same.
If the sum of proper divisors turns out to exactly match the original number, something genuinely curious happens. 6's proper divisors are 1, 2, 3, and adding them up gives 1+2+3=6 — exactly 6! A number like this is called a perfect number. The next perfect number after 6 is 28 (1+2+4+7+14=28), followed by 496, then 8128. Every even perfect number follows the exact pattern 2ⁿ⁻¹×(2ⁿ−1) (as long as 2ⁿ−1 is prime), and the rule for these has been fully worked out — but whether an odd perfect number exists at all is a problem no one has managed to prove in over 2,000 years.
If a perfect number is "a loop that returns to itself," some pairs of numbers form a matched pair that points at each other instead. Add up all of A's proper divisors and you get B; add up all of B's proper divisors and you get A right back. The smallest example is 220 and 284. Add up 220's proper divisors and you get 284; add up 284's proper divisors and you get exactly 220. A pair like this is called amicable numbers, and the concept is old enough to have been known since ancient Greek times.
Sometimes the loop stretches beyond just two numbers — to three, five, or even more — and this is called sociable numbers. For example, starting from 12496 and finding the sum of its proper divisors gives 14288; do it again and you get 15472, then 14536, then 14264, before finally returning to 12496 — a loop of 5 numbers. Interestingly, not a single sociable chain of exactly length 3 has ever been found — whether it truly doesn't exist, or simply hasn't been found yet, remains an unsolved problem.
A case that works out this perfectly is extremely rare. Most natural numbers are either deficient, where the sum of proper divisors is less than the number itself (like 8, whose proper-divisor sum is 1+2+4=7), or abundant, where the sum is greater (like 12, whose proper-divisor sum is 1+2+3+4+6=16). A perfect number sits exactly balanced between the two — an extremely rare borderline case.
On our activity page, you can check perfect numbers, amicable numbers, and sociable numbers one at a time with preset buttons, and follow a sociable chain step by step with a slider. In the "classify deficient, perfect, or abundant" section, you can pick any number from 2 to 220 and calculate the sum of its proper divisors yourself to see which category it falls into. Finish up with the quiz to check what you've learned.