Abundant number

In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the sum-of-divisors function: the sum of all positive divisors of n, including n itself. The value σ(n) − 2n is called the abundance of n. An equivalent definition is that the proper divisors of the number (the divisors except the number itself) sum to more than the number.

The first few abundant numbers (sequence A005101 in OEIS) are:

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, …

As an example, consider the number 24. Its divisors are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number 24 is abundant. Its abundance is 60 − 2 × 24 = 12.

The smallest abundant number not divisible by two, i.e. odd, is 945, and the smallest not divisible by 2 or by 3 is 5391411025 whose prime factors are 5², 7, 11, 13, 17, 19, 23, and 29. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If A(k) represents the smallest abundant number not divisible by the first k primes then for all ε > 0 we have:

(1 − ε)(klnk)^{2 − ε} < lnA(k) < (1 + ε)(klnk)^{2 + ε} for k sufficiently large.

Infinitely many even and odd abundant numbers exist. Marc Deléglise showed in 1998 that the natural density of abundant numbers is between 0.2474 and 0.2480. 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.

Closely related to abundant numbers are perfect numbers with σ(n) = 2n, and deficient numbers with σ(n) < 2n. The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100).

External links

• The Prime Glossary: Abundant number
• Weisstein, Eric W., "Abundant Number" from MathWorld.
• abundant number at PlanetMath.

References

• M. Deléglise, "Bounds for the density of abundant integers," Experimental Math., 7:2 (1998), pp. 137-143.
• D. Iannucci, "On the smallest abundant number not divisible by the first k primes" Bull. Belgian Math. Soc., 12 (2005), pp. 39–44.