Amicable Pair Finder

How it is calculated

The tool needs a whole number from 1 to 1,000,000,000,000. A decimal such as 220.4, zero, a negative number or anything larger shows no result. It computes s(n), the sum of the proper divisors of n (every divisor except n itself), and shows it as the partner. To test the pair it computes s(s(n)) and answers Yes only when the partner is a different number and s(partner) = n. Divisors are found by trial division up to the square root of n, pairing each small divisor d with n ÷ d, so very large inputs take longer. The partner is always shown, even when the answer is No, and is then just the divisor sum. A perfect number such as 6 equals its own sum, so it is reported as No. The number 1 has no proper divisors and gives 0.

Common mistakes

• Reading the partner box as proof of an amicable pair. It only shows the divisor sum: 12 shows 16, yet the answer is No because the proper divisors of 16 add up to 15, not 12.
• Including the number itself among its divisors. The sum is over proper divisors only, which is why 6 gives 6 and not 12.
• Calling a perfect number amicable. 6 and 28 equal their own divisor sum, and the tool answers No because an amicable pair needs two different numbers.
• Typing a decimal like 220.4 and expecting it to be rounded. Only whole numbers are accepted, so no result appears until you enter 220.
• Assuming the pair can only be found from its smaller member. It works from both: entering 284 gives partner 220 and Yes.