Abundant number

An abundant number is an integer whose proper divisors sum to more than the number itself; contrasted with deficient and perfect numbers and related to divisor-sum functions and special classes like weird numbers.

Overview

An abundant number is a positive integer n for which the sum of its divisors exceeds 2n, or equivalently the sum of its proper divisors is greater than n. That condition measures whether a number carries "extra" divisors beyond itself. Abundant numbers form one of three basic classes of natural numbers under divisor-sum comparisons; the other two are deficient or perfect numbers.

Definition and notation

Let σ(n) denote the sum of divisors function, which adds every positive divisor of n including 1 and n. Then n is abundant when σ(n) > 2n. Equivalently, if s(n) = σ(n) − n is the sum of proper divisors of n, the abundance condition is s(n) > n. The quantity s(n) − n is sometimes called the abundance of n and can be used to compare how strongly abundant one number is relative to another.

Examples and small cases

The smallest abundant number is 12, because its proper divisors 1, 2, 3, 4, and 6 sum to 16, which is greater than 12. Other small abundant numbers include 18, 20, 24 and 30; for instance 18 has proper divisors 1, 2, 3, 6 and 9 which add to 21. The first odd abundant number is 945, so odd abundant numbers are comparatively rare among small integers.

12: 1+2+3+4+6 = 16 > 12
18: 1+2+3+6+9 = 21 > 18
20: 1+2+4+5+10 = 22 > 20

Abundant numbers are closely tied to other divisor-based concepts. A semiperfect (or pseudoperfect) number is one that equals the sum of some of its proper divisors; many abundant numbers are semiperfect, but some are not. A weird number is abundant yet not semiperfect; 70 is the smallest known weird number. A primitive abundant number is abundant while none of its proper divisors is abundant.

Studying abundance intersects multiplicative number theory because σ(n) is multiplicative for coprime arguments, which helps analyze abundant numbers composed from prime powers. Patterns differ between even and odd integers: even abundant numbers are common, while odd abundant numbers require more intricate combinations of prime powers and are less frequent in small ranges.

History, sequences and significance

The classification of numbers by divisor sums has roots in ancient mathematics, where perfect numbers drew much attention. Modern study uses functions like σ(n) to classify abundant and deficient numbers and to examine their distribution. Lists and sequences of abundant numbers appear in integer-sequence collections (for example in standard sequence catalogs). Researchers examine their density, the existence of primitive and weird examples, and their role in additive problems that ask which integers can be written as sums of abundant numbers.