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 2*n*; the value *σ(n)* - 2*n* 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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- M. Deléglise, "Bounds for the density of abundant integers,"
*Experimental Math.,*7:2 (1998) p. 137-143.