Deficient Number Checker

About this calculator

A whole number is deficient when the sum of its proper divisors — all divisors excluding the number itself — is less than the number. For example, 8's proper divisors are 1, 2, and 4, which sum to 7, less than 8, making it deficient; the difference (1, here) is called its deficiency.

This classification, going back to ancient Greek number theory (Euclid and Nicomachus), splits all positive integers into three camps: deficient (divisor sum less than n), perfect (divisor sum exactly n, like 6 = 1+2+3 or 28), and abundant (divisor sum greater than n, like 12 = 1+2+3+4+6 = 16). Most numbers are actually deficient — all prime numbers are, since their only proper divisor is 1 — which is why perfect numbers are famously rare (only 51 are known as of the largest Mersenne-prime searches) and abundant numbers, while more common than perfect ones, are still a minority among the integers.