# Abundant number

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. 12 has an abundance of 4, for example.

## Definition

A number n for which the sum of divisors σ(n)>2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)>n.

Abundance is the value σ(n)-2n (or s(n)-n).

## Examples

The first few abundant numbers are:

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

For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is more than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12.

## Properties

• The smallest odd abundant number is 945
• The smallest not divisible by 2 or by 3 is 5391411025 whose prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence in OEIS). An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes.[1] If $A(k)$ represents the smallest abundant number not divisible by the first k primes then for all $\epsilon>0$ we have:
$(1-\epsilon)(k\ln k)^{2-\epsilon}<\ln A(k)<(1+\epsilon)(k\ln k)^{2+\epsilon}$ for k sufficiently large.
• Infinitely many even and odd abundant numbers exist. Marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480.[2]
• Every proper multiple of a perfect number, and every multiple of an abundant number, is abundant.
• Every integer greater than 20161 can be written as the sum of two abundant numbers.[3]
• 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, however, none have yet been found.

## Related concepts

Closely related to abundant numbers are perfect numbers, that is numbers the sum of whose proper factors equals the number itself (such as 6 and 28) (or more formally, σ(n) = 2n), and deficient numbers, or numbers the sum of whose proper factors is less than the number itself (or σ(n) < 2n.)

The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100) who described abundant numbers as like deformed animals with too many limbs.

The abundancy index of n is the ratio σ(n)/n.[4]

The sequence (ak) of least numbers n such that σ(n) > kn, in which a2 = 12 corresponds to the first abundant number, grows extremely quickly (sequence in OEIS).

If p = (p1,...,pn) is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant. A necessary and sufficient condition for this is that the product of pi/(pi-1) be at least 2.[5]

## References

1. D. Iannucci (2005), "On the smallest abundant number not divisible by the first k primes", Bulletin of the Belgian Mathematical Society 12 (1): 39–44
2. M. Deléglise (1998). "Bounds for the density of abundant integers". Experimental Mathematics 7 (2): 137–143. MR 1677091.
3. "Sloane's A048242 : Numbers that are not the sum of two abundant numbers", . OEIS Foundation.
4. Laatsch, Richard (1986). "Measuring the abundancy of integers". 59 (2): 84–92. ISSN 0025-570X. JSTOR 2690424. MR 0835144. Zbl 0601.10003.
5. Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". 44 (3): 328–339. doi:10.1006/jnth.1993.1057. MR 1233293. Zbl 0781.11015.
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