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[[File:Highly abundant number Cuisenaire rods 8.png|thumb|Sums of the divisors, in [[Cuisenaire rod]]s, of the first six highly abundant numbers]]

In [[mathematics]], a '''highly abundant number''' is a [[natural number]] with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.

Highly abundant numbers and several similar classes of numbers were first introduced by {{harvs|authorlink=Subbayya Sivasankaranarayana Pillai|last=Pillai|year=1943|txt}}, and early work on the subject was done by {{harvs|author1-link=Leonidas Alaoglu|last1=Alaoglu|author2-link=Paul Erdős|last2=Erdős|year=1944|txt}}. Alaoglu and Erdős tabulated all highly abundant numbers up to 10<sup>4</sup>, and showed that the number of highly abundant numbers less than any ''N'' is at least proportional to log<sup>2</sup> ''N''.

== Formal definition and examples ==

Formally, a natural number ''n'' is called highly abundant [[if and only if]] for all natural numbers ''m'' &lt; ''n'',

:<math>\sigma(n) > \sigma(m)</math>

where σ denotes the [[divisor function|sum-of-divisors function]]. The first few highly abundant numbers are
:[[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], [[8 (number)|8]], [[10 (number)|10]], [[12 (number)|12]], [[16 (number)|16]], [[18 (number)|18]], [[20 (number)|20]], [[24 (number)|24]], [[30 (number)|30]], [[36 (number)|36]], [[42 (number)|42]], [[48 (number)|48]], [[60 (number)|60]], ... {{OEIS|id=A002093}}.

For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.

The only odd highly abundant numbers are 1 and 3.<ref>See {{harvtxt|Alaoglu|Erdős|1944}}, p.&nbsp;466. Alaoglu and Erdős claim more strongly that all highly abundant numbers greater than 210 are divisible by 4, but this is not true: 630 is highly abundant, and is not divisible by 4. (In fact, 630 is the only [[counterexample]]; all larger highly abundant numbers are divisible by 12.)</ref>

== Relations with other sets of numbers ==
{{Euler_diagram_numbers_with_many_divisors.svg}}
Although the first eight [[factorial]]s are highly abundant, not all factorials are highly abundant. For example,
:&sigma;(9!) = &sigma;(362880) = 1481040,
but there is a smaller number with larger sum of divisors,
:&sigma;(360360) = 1572480,
so 9! is not highly abundant.

Alaoglu and Erdős noted that all [[superabundant number]]s are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by {{harvs|authorlink=Jean-Louis Nicolas|first=Jean-Louis|last=Nicolas|year=1969|txt}}.

Despite the terminology, not all highly abundant numbers are [[abundant number]]s.  In particular, none of the first seven highly abundant numbers is abundant.

7200 is the largest [[powerful number]] that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once. Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors.<ref>{{harvtxt|Alaoglu|Erdős|1944}}, pp. 464–466.</ref>

==Notes==
{{reflist}}
== References ==
*{{cite journal
 | last1 = Alaoglu | first1 = L. | author1-link = Leonidas Alaoglu
 | last2 = Erdős | first2 = P. | author2-link = Paul Erdős
 | title = On highly composite and similar numbers
 | journal = [[Transactions of the American Mathematical Society]]
 | volume = 56
 | year = 1944
 | pages = 448–469
 | mr = 0011087 
 | doi = 10.2307/1990319
 | issue = 3
 | jstor = 1990319
 | url = http://combinatorica.hu/~p_erdos/1944-03.pdf
 | ref = harv}}
*{{cite journal
 | last = Nicolas | first = Jean-Louis | authorlink = Jean-Louis Nicolas
 | title = Ordre maximal d'un élément du groupe ''S<sub>n</sub>'' des permutations et "highly composite numbers"
 | journal = Bull. Soc. Math. France
 | volume = 97
 | year = 1969
 | pages = 129–191
 | mr = 0254130 
 | url = http://www.numdam.org/item?id=BSMF_1969__97__129_0
 | ref = harv}}
*{{cite journal
 | last = Pillai | first = S. S.
 | authorlink = Subbayya Sivasankaranarayana Pillai
 | title = Highly abundant numbers
 | journal = Bull. Calcutta Math. Soc.
 | volume = 35
 | year = 1943
 | pages = 141–156
 | mr = 0010560
 | ref = harv}}

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

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