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# Abundant number

In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the divisor function: the sum of all positive divisors of n, including n itself. The value σ(n) − 2n is called the abundance of n.

Abundant numbers were first introduced in Nicomachus' Introductio Arithmetica (circa 100). He referred to them as superabundant numbers, and only required that σ(n) exceeds n.

The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (sequence A005101 in OEIS). The first odd abundant number is 945. 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 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.

## References

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