abundant number

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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...

, an abundant number or excessive number is a number for which the sum of its
proper divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s is greater than the number. 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'') > 2''n'', or, equivalently, the sum of proper divisors (or
aliquot sum In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or ...
) ''s''(''n'') > ''n''. Abundance is the value ''σ''(''n'') − ''2n'' (or ''s''(''n'') − ''n'').

# Examples

The first 28 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, ... . For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is greater 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 factor A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s are 5, 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''
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...

s. If $A\left(k\right)$ represents the smallest abundant number not divisible by the first ''k'' primes then for all $\epsilon>0$ we have ::$\left(1-\epsilon\right)\left(k\ln k\right)^<\ln A\left(k\right)<\left(1+\epsilon\right)\left(k\ln k\right)^$ :for sufficiently large ''k''. *Every multiple of a
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
(except the perfect number itself) is abundant.Tattersall (2005) p.134 For example, every multiple of 6 greater than 6 is abundant because $1 + \tfrac + \tfrac + \tfrac = n +1.$ *Every multiple of an abundant number is abundant. For example, every multiple of 20 (including 20 itself) is abundant because $\tfrac + \tfrac + \tfrac + \tfrac + \tfrac= n + \tfrac.$ * Consequently, infinitely many even and odd abundant numbers exist. *Furthermore, the set of abundant numbers has a non-zero
natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the Set (mathematics), set of natural numbers is. It relies chiefly on the probability of encountering ...
. 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. * An abundant number which is not the multiple of an abundant number or perfect number (i.e. all its proper divisors are deficient) is called a primitive abundant number * An abundant number whose abundance is greater than any lower number is called a highly abundant number, and one whose relative abundance (i.e. s(n)/n ) is greater than any lower number is called a
superabundant number In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number ''n'' is called superabundant precisely when, for all ''m'' < ''n'' :\frac 6/5. Superabundant numbers were defined by . ...
*Every
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
greater than 20161 can be written as the sum of two abundant numbers. *An abundant number which is not a
semiperfect number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or ...
is called a
weird number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or ...
.Tattersall (2005) p.144 An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found. *Every abundant number is a multiple of either a perfect number or a primitive abundant number.

# Related concepts

Numbers whose sum of proper factors equals the number itself (such as 6 and 28) are called
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...
s, while numbers whose sum of proper factors is less than the number itself are called
deficient number In number theory, a deficient number or defective number is a number ''n'' for which the Divisor function#Definition, sum of divisors of ''n'' is less than 2''n''. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum ...
s. The first known classification of numbers as deficient, perfect or abundant was by
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Ancient Greek, Greek. ...
in his '' Introductio Arithmetica'' (circa 100 AD), which described abundant numbers as like deformed animals with too many limbs. The abundancy index of ''n'' is the ratio ''σ''(''n'')/''n''. Distinct numbers ''n''1, ''n''2, ... (whether abundant or not) with the same abundancy index are called friendly numbers. The sequence (''a''''k'') of least numbers ''n'' such that ''σ''(''n'') > ''kn'', in which ''a''2 = 12 corresponds to the first abundant number, grows very quickly . The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 33 × 52 × 72 × 11 × 13 × 17 × 19 × 23 × 29. If p = (''p''1, ..., ''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 > 2.

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