Perfect Number Calculator

Formula

How this is calculated

The single input n must be a positive integer. The calculator finds every proper divisor of n — that is, every positive divisor except n itself — and adds them together. This total is called the aliquot sum, written s.

To find the divisors efficiently, it loops i from 1 up to floor(sqrt(n)). Whenever n is divisible by i, both i and the paired divisor n/i are divisors, so both are added (taking care to add a square root only once when n is a perfect square, e.g. for n = 16 the divisor 4 is counted a single time). The divisor n itself is excluded because we only sum proper divisors.

The number is then classified by comparing s with n: it is perfect when s = n (for example 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14), abundant when s > n (such as 12), and deficient when s < n (such as 8, and every prime, whose only proper divisor is 1). The abundance, s − n, measures how far the number is from being perfect: zero for perfect numbers, positive for abundant, negative for deficient. By convention n = 1 has no proper divisors so s = 0 and it is deficient.