Abundant number

In number theory, an abundant number or excessive number is a number n for which the sum of divisors σ(n)>2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)>n. The value σ(n)-2n (or s(n)-n) is known as the abundance.

Examples

The first few abundant numbers are:

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, … (sequence A005101 in the OEIS).

For example, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number 24 is abundant. Its abundance is 60 − (2 × 24) = 12.

Properties

• The smallest odd abundant number is 945

• The smallest not divisible by 2 or by 3 is 5391411025 whose prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence A047802 in the OEIS). An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If $A(k)$ represents the smallest abundant number not divisible by the first k primes then for all $\epsilon >0$ we have:

$(1-\epsilon )(k\ln k)^{2-\epsilon }<\ln A(k)<(1+\epsilon )(k\ln k)^{2+\epsilon }$ for k sufficiently large.

• Infinitely many even and odd abundant numbers exist. Marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480.

• Every proper multiple of a perfect number, and every multiple of an abundant number, is abundant.

• Every integer greater than 20161 can be written as the sum of two abundant numbers.

• An abundant number which is not a semiperfect number is called a weird number; an abundant number with abundance 1 is called a quasiperfect number, however, none have yet been found.

Related concepts

Closely related to abundant numbers are perfect numbers, that is numbers the sum of whose proper factors equals the number itself (such as 6 and 28) (or more formally, σ(n) = 2n), and deficient numbers, or numbers the sum of whose proper factors is less than the number itself (or σ(n) < 2n.) The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100) who described abundant numbers as like deformed animals with too many limbs.

External links

• The Prime Glossary: Abundant number
• Weisstein, Eric W. "Abundant Number". MathWorld.
• abundant number at PlanetMath.