Editing Abundant number (section)

==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.