Abundant / Perfect / Deficient Number Checker

How it is calculated

The tool adds up all proper divisors of n, meaning every positive divisor smaller than n itself (the aliquot sum s(n)), and compares it with n. If s(n) > n the number is Abundant, if s(n) = n it is Perfect, and if s(n) < n it is Deficient. It finds divisors in pairs by testing every i up to √n, so even large numbers are quick.

The input is treated as a whole number between 1 and 100,000,000; anything outside this range, such as 0, a negative number or 100,000,001, gives no result. Decimals are cut to the integer part. The number 1 has no proper divisors, so its aliquot sum is 0 and it is reported as Deficient.

Common mistakes

• Including the number itself in the divisor sum; for 6 that would give 12 and hide that 6 is perfect (1 + 2 + 3 = 6).
• Mixing this up with the sigma function σ(n): σ(n) counts n as well, so a number is perfect when σ(n) = 2n, while this tool compares the sum without n to n.
• Assuming even numbers are always abundant or odd ones always deficient; 12 is abundant but 945 is the smallest odd abundant number, and 8 is an even deficient number.
• Entering a decimal such as 28.9, which is cut to 28 rather than rounded.
• Entering numbers above 100,000,000 and wondering why nothing appears; the tool has a hard upper limit.