Editing Abundant number

{{short description|Number that is less than the sum of its proper divisors}}
[[File:Abundant number Cuisenaire rods 12.png|thumb|275px|Demonstration, with [[Cuisenaire rods]], of the abundance of the number 12]]

In [[number theory]], an '''abundant number''' or '''excessive number''' is a [[positive integer]] for which the sum of its [[proper divisor]]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==
An ''abundant number'' is a [[natural number]] {{math|''n''}} for which the [[Divisor function|sum of divisors]] {{math|''σ''(''n'')}} satisfies {{math|''σ''(''n'') > 2''n''}}, or, equivalently, the sum of proper divisors (or [[aliquot sum]]) {{math|''s''(''n'')}} satisfies {{math|''s''(''n'') > ''n''}}.

The ''abundance'' of a natural number  is the [[integer]] {{math|''σ''(''n'') − ''2n''}} (equivalently, {{math|''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, ... {{OEIS|id=A005101}}.

For example, the proper divisors of 24 are 1,&nbsp;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&nbsp;−&nbsp;24&nbsp;=&nbsp;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]]s are 5, 7, 11, 13, 17, 19, 23, and 29 {{OEIS|id=A047802}}. An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first ''k'' [[Prime number|prime]]s.<ref>{{citation |author=D. Iannucci |title=On the smallest abundant number not divisible by the first ''k'' primes |journal=[[Bulletin of the Belgian Mathematical Society]] |volume=12 |issue=1 |pages=39–44 |year=2005 |doi=10.36045/bbms/1113318127 |url=https://projecteuclid.org/journals/bulletin-of-the-belgian-mathematical-society-simon-stevin/volume-12/issue-1/On-the-smallest-abundant-number-not-divisible-by-the-first/10.36045/bbms/1113318127.full}}</ref> If <math>A(k)</math> represents the smallest abundant number not divisible by the first ''k'' primes then for all <math>\epsilon>0</math> we have
::<math> (1-\epsilon)(k\ln k)^{2-\epsilon}<\ln A(k)<(1+\epsilon)(k\ln k)^{2+\epsilon} </math> 
:for sufficiently large ''k''.
*Every multiple of a [[perfect number]] (except the perfect number itself) is abundant.<ref name=Tat134>Tattersall (2005) p.134</ref>  For example, every multiple of 6 greater than 6 is abundant because <math>1 + \tfrac{n}{2} + \tfrac{n}{3} + \tfrac{n}{6} = n +1.</math>
*Every multiple of an abundant number is abundant.<ref name=Tat134/>  For example, every multiple of 20 (including 20 itself) is abundant because <math>\tfrac{n}{2} + \tfrac{n}{4} + \tfrac{n}{5} + \tfrac{n}{10} + \tfrac{n}{20}= n + \tfrac{n}{10}.</math>
* Consequently, infinitely many [[Even and odd numbers|even and odd]] abundant numbers exist.
[[File:Proportion of abundant numbers.svg|thumb|Let <math>a(n)</math> be the number of abundant numbers not exceeding <math>n</math>. Plot of <math>a(n)/n</math> for <math>n < 10^6</math> (with <math>n</math>  log-scaled)]]
*Furthermore, the set of abundant numbers has a non-zero [[natural density]].<ref name=HT95>{{cite book | zbl=0653.10001 | last1=Hall | first1=Richard R. | last2= Tenenbaum | first2=Gérald | author2-link=Gérald Tenenbaum | title=Divisors | series=Cambridge Tracts in Mathematics | volume=90 | location =Cambridge | publisher=[[Cambridge University Press]] | year=1988 | isbn=978-0-521-34056-4 | page=95 }}</ref>  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.<ref name=Del1998>{{cite journal | first=Marc | last=Deléglise | title= Bounds for the density of abundant integers | journal=Experimental Mathematics | volume=7 | issue=2 | year=1998 | pages=137–143 | url= http://projecteuclid.org/euclid.em/1048515661 | mr=1677091 | zbl=0923.11127 | issn=1058-6458 | doi=10.1080/10586458.1998.10504363| citeseerx = 10.1.1.36.8272 }}</ref>
* 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]]
*Every [[integer]] greater than 20161 can be written as the sum of two abundant numbers. The largest even number that is not the sum of two abundant numbers is 46.<ref>{{Cite OEIS|sequencenumber=A048242|name=Numbers that are not the sum of two abundant numbers}}</ref>
*An abundant number which is not a [[semiperfect number]] is called a [[weird number]].<ref name=Tat144>Tattersall (2005) p.144</ref>  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==
{{Euler_diagram_numbers_with_many_divisors.svg}}
Numbers whose sum of proper factors equals the number itself (such as 6 and 28) are called [[perfect number]]s, while numbers whose sum of proper factors is less than the number itself are called [[deficient number]]s. The first known classification of numbers as deficient, perfect or abundant was by [[Nicomachus]] in his ''[[Introduction to Arithmetic|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''.<ref>{{cite journal | last=Laatsch | first=Richard | title=Measuring the abundancy of integers | journal=[[Mathematics Magazine]] | volume=59 | number=2 | pages=84–92 | year=1986 | issn=0025-570X | zbl=0601.10003 |jstor=2690424 |mr=0835144 | doi=10.2307/2690424}}</ref> Distinct numbers ''n''<sub>1</sub>, ''n''<sub>2</sub>, ... (whether abundant or not) with the same abundancy index are called [[friendly number]]s.

The sequence (''a''<sub>''k''</sub>) of least numbers ''n'' such that ''σ''(''n'') > ''kn'', in which ''a''<sub>2</sub> = 12 corresponds to the first abundant number, grows very quickly {{OEIS|id=A134716}}.

The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 3<sup>3</sup> × 5<sup>2</sup> × 7<sup>2</sup> × 11 × 13 × 17 × 19 × 23 × 29.<ref>For smallest odd integer ''k'' with abundancy index exceeding ''n'', see {{Cite OEIS|sequencenumber=A119240|name=Least odd number ''k'' such that sigma(k)/k >= n.}}</ref>

If '''p''' = (''p''<sub>1</sub>, ..., ''p<sub>n</sub>'') 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<sub>i</sub>''/(''p<sub>i</sub>'' − 1) be > 2.<ref>{{cite journal | title = Sums of divisors and Egyptian fractions | last = Friedman | first = Charles N. | journal = [[Journal of Number Theory]] | year = 1993 | volume = 44 | pages = 328–339 | mr = 1233293 | zbl = 0781.11015 | doi = 10.1006/jnth.1993.1057 | issue = 3 | doi-access = free }}</ref>

== References ==
<references/>
* {{cite book | title=Elementary Number Theory in Nine Chapters | first=James J. | last=Tattersall | edition=2nd | publisher=[[Cambridge University Press]] | year=2005 | isbn=978-0-521-85014-8 | zbl=1071.11002 }}

== External links ==
* [http://primes.utm.edu/glossary/page.php?sort=AbundantNumber The Prime Glossary: Abundant number]
* {{MathWorld |urlname=AbundantNumber |title=Abundant Number}}
* {{PlanetMath |urlname=AbundantNumber |title=Abundant number |id=7869}}

{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Arithmetic dynamics]]
[[Category:Divisor function]]
[[Category:Integer sequences]]