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. The integer 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. The number 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, 104, 108, 112, 114, 120, … (sequence A005101 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 abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence A047802 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.
The set of abundant numbers has a natural density.[2] 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.[3]
Every proper multiple of a perfect number is abundant.[4] For example, every proper multiple of 6 is abundant because n/2 + n/3 + n/6 = n, and if a proper multiple, that provides an extra divisor.
Every multiple of an abundant number is abundant.[4] For example, every multiple of 20 (including 20 itself) is abundant because n/2 + n/4 + n/5 + n/10 + n/20 = n + n/10.
Every integer greater than 20161 can be written as the sum of two abundant numbers.[5]
An abundant number which is not a semiperfect number is called a weird number.[6] An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found.

Related concepts

Closely related to abundant numbers are perfect numbers. These are numbers whose sum of proper factors equals the number itself (such as 6 and 28) (or more formally, σ(n) = 2n), and deficient numbers, or numbers whose sum of 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.[7] Distinct numbers n1, n2, ... (whether abundant or not) with the same abundancy index are called friendly numbers.

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 A134716 in OEIS).

If p = (p₁,...,p_n) 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 p_i/(p_i-1) be at least 2.^[8]

References

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
Hall, Richard R.; Tenenbaum, Gérald (1988). Divisors. Cambridge Tracts in Mathematics 90. Cambridge: Cambridge University Press. p. 95. ISBN 0-521-34056-X. Zbl 0653.10001.
Deléglise, Marc (1998). "Bounds for the density of abundant integers". Experimental Mathematics 7 (2): 137–143. doi:10.1080/10586458.1998.10504363. ISSN 1058-6458. MR 1677091. Zbl 0923.11127.
Tattersall (2005) p.134
"Sloane's A048242 : Numbers that are not the sum of two abundant numbers", The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Tatersall (2005) p.144
Laatsch, Richard (1986). "Measuring the abundancy of integers". Mathematics Magazine 59 (2): 84–92. ISSN 0025-570X. JSTOR 2690424. MR 0835144. Zbl 0601.10003.

Friedman, Charles N. (1993). "Sums of divisors and Egyptian fractions". Journal of Number Theory 44 (3): 328–339. doi:10.1006/jnth.1993.1057. MR 1233293. Zbl 0781.11015.

Tattersall, James J. (2005). Elementary Number Theory in Nine Chapters (2nd ed.). Cambridge University Press. ISBN 0-521-85014-2. Zbl 1071.11002.

External links

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

Mathematics Encyclopedia