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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.
The abundancy index of n is the ratio σ(n)/n.
The sequence (ak) of least numbers n such that σ(n) > kn, in which a2 = 12 corresponds to the first abundant number, grows extremely quickly (sequence A134716 in OEIS).
If p = (p1,...,pn) is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant. A necessary and sufficient condition for this is that the product of pi/(pi-1) be at least 2.
Read more about this topic: Abundant Number
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