In mathematics, an abundant number (also sometimes called an excessive number) is a number n for which the sum of all its positive divisors (including n, the divisor function, σ(n)) is greater than 2n; the value σ(n) - 2n is sometimes called the abundance of n. Abundant numbers were first introduced in Nicomachus' Introductio Arithmetica (circa 100).
The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (Sloane's A005101); M. Deléglise showed in 1998 that the natural density of abundant numbers is in the open interval ]0.2474, 0.2480[.
An infinite number of both even and odd abundant numbers exist (for example, all multiples of 12 and all odd multiples 945 are abundant); furthermore, every proper multiple of a perfect number and every multiple of an abundant number is abundant. Also, every integer >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.
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