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. The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... . The first odd abundant number is 945. Marc Deléglise showed in 1998 that the natural density of abundant numbers is in the open interval (0.2474, 0.2480).
Infinitely many even and odd abundant numbers exist. 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.
- M. Deléglise, "Bounds for the density of abundant integers," Experimental Math., 7:2 (1998) p. 137-143.
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