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In [[number theory]], an '''abundant number''' or '''excessive number''' is a number for which the sum of its [[proper divisor]]s 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.
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==Definition==
A number ''n'' for which the sum of divisors [[Divisor function|''&sigma;''(''n'')]]>2''n'', or, equivalently, the sum of proper divisors (or [[aliquot sum]]) ''s''(''n'')>''n''.
 
Abundance is the value ''&sigma;''(''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, … {{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 more 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 prime factors 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'' primes.<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|year=2005|pages=39–44|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bbms/1113318127}}</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 ''k'' sufficiently large.
 
*Infinitely many [[Even and odd numbers|even and odd]] abundant numbers exist.
*The set of abundant numbers has a [[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=0-521-34056-X | 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 }}</ref>
*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.<ref>{{SloanesRef |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]]. 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 number]]s, that is numbers the sum of whose proper factors equals the number itself (such as 6 and 28) (or more formally, ''σ''(''n'')&nbsp;=&nbsp;2''n''), and [[deficient number]]s, or numbers the sum of whose proper factors is less than the number itself (or ''σ''(''n'')&nbsp;<&nbsp;2''n''.)
 
The [[natural number]]s were first classified as either deficient, perfect or abundant by [[Nicomachus]] in his ''[[Introduction to Arithmetic|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''.<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 }}</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 extremely quickly {{OEIS|id=A134716}}.
 
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 at least 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 | url=http://dell5.ma.utexas.edu/users/friedman/divisors.ps | mr= 1233293 | zbl=0781.11015 | doi = 10.1006/jnth.1993.1057 | issue = 3}}</ref>
 
== References ==
<references/>
 
== 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:Divisor function]]
[[Category:Integer sequences]]

Latest revision as of 23:40, 9 December 2014

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Intown, Newport Rhode Island Romantic Victorian Inn. This five room inn is located in the heart of Newport. The harbor, beaches, Cliff Walk, mansions, restaurants and shopping are all within walking distance. Each room is individually designed for your comfort and pleasure. Friendly hosts. It features 14 large, airy guest rooms (many with fireplaces), 11 foot ceilings, sweeping lawns and historic specimen trees. It has been lovingly restored and awaits your return to this golden era. Come on a romantic weekend getaway in a room with a fireplace or plan a weeklong stay with the family and see all that our area has to offer. Close to Rhode Island's world famous beaches, historic Newport, Mystic Seaport, the Block Island.A dramatic seaside walk offering glimpses of the magnificent Newport mansions. The first half follows Ocean Drive along the magnificent shoreline jutting out from Newport. It also has the area's most spectacular ocean vistas, fronting oceanswept cliffs. The Norman Bird Sanctuary maintains 8 miles of woodland trails close to the ocean's edge.

Prevention is the name of the game when you are in your 20s. Sun protection is essential, so ensure you wear an SPF moisturiser. "It is also a good idea to invest in a good cleansing routine and you can start to exfoliate regularly to avoid skin congestion," recommends Jemma. Night cream isn't necessary, as your epidermal cells are regenerating every 20 days. It may be www.stevensassoc.co.uk/michaelkors/outlet.php?p=michael+kors+handbags+outlet wise to invest www.stevensassoc.co.uk/michaelkors/outlet.php?p=michael+kors+handbags+outlet+uk in a good tinted moisturiser and a concealer to disguise undereye circles from late nights! And, of course, drink plenty of water and eat a balanced, healthy diet (as recommended by the Weight Watchers Guidelines).

Wardlaw Building: Beginning this weekend, the former home of the Northwest Georgia Arts Guild, 309 N. Main St., will be open as a LaFayettearea welcome center. Its volunteer staff also will offer tours of the Marsh House and Chattooga Academy. The exhibit on view centers on the Civil War in Walker County.

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