The Abundant number reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

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.

Table of contents

1 See also 2 References 3 External links

See also

• Deficient number
• Perfect number

References

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

External links

• The Prime Glossary: Abundant number
• MathWorld: Abundant number

This is the "Abundant number" reference article from the English Wikipedia. All text is available under the terms of the GNU Free Documentation License. See also our Disclaimer.