Sum of Divisors Calculator

The sum of divisors of a whole number is the total of every positive whole number that divides it exactly, including 1 and the number itself. For 12 the divisors are 1, 2, 3, 4, 6 and 12, which add up to 28. Mathematicians call this the sigma function, written with the Greek letter sigma.

The aliquot sum is the sum of a number's proper divisors, that is, all its divisors except the number itself. It equals the full sum of divisors minus the number. For 12 the aliquot sum is 28 minus 12, which equals 16. The aliquot sum tells you whether a number is deficient, perfect, or abundant.

A perfect number equals the sum of its proper divisors, so its aliquot sum equals the number itself. The smallest perfect number is 6, since 1 plus 2 plus 3 equals 6. The next is 28. This calculator reports whether your input is perfect, deficient (aliquot sum less than the number), or abundant (aliquot sum greater).

The calculator tests divisors only up to the square root of the number. Each divisor i found below the root contributes both i and the paired quotient N divided by i to the sum, with care taken not to count the square root twice for a perfect square. This is far faster than checking every number up to N.

The sum of divisors is defined for positive whole numbers. The calculator floors any decimal you enter and rejects zero and negatives. To keep the page fast, very large inputs are capped, and the cap is shown if you exceed it. For a prime number the sum of divisors is simply the number plus one.