Perfect Number Checker

How it is calculated

The tool adds up all the proper divisors of n, meaning every divisor except n itself, and compares that sum s(n) with n. If s(n) = n the number is perfect; if s(n) > n it is abundant; if s(n) < n it is deficient. Divisors are found in pairs by testing every i up to the square root of n, so large values are quick. n must be a whole number from 1 to 100,000,000; decimals are cut to the integer part and values outside the range give no result. The number 1 has no proper divisors, so its sum is 0 and it is deficient.

Common mistakes

• Including the number itself in the divisor sum; then every perfect number would look like double (28 gives 56).
• Leaving out 1, which is a proper divisor of every number.
• Thinking that abundant means large or deficient means small; 945 is odd and abundant, while a prime is always deficient.
• Entering a decimal like 28.9 and expecting it to be treated as 29; it is truncated to 28.
• Expecting large perfect numbers by brute force; they are very rare, and the tool stops at 100,000,000.