View source for Deficient number

{{short description|Number whose divisor sum is less than itself}}
[[File:Deficient number Cuisenaire rods 8.png|thumb|Demonstration, with [[Cuisenaire rods]], of the deficiency of the number 8]]

In [[number theory]], a '''deficient number''' or '''defective number''' is a positive [[integer]] {{mvar|n}} for which the [[Divisor function#Definition|sum of divisors]] of {{mvar|n}} is less than {{math|2''n''}}.  Equivalently, it is a number for which the sum of [[proper divisor]]s (or [[aliquot sum]]) is less than {{mvar|n}}.  For example, the proper divisors of 8 are {{nowrap|1, 2, and 4}}, and their sum is less than 8, so 8 is deficient.

Denoting by {{math|''σ''(''n'')}} the sum of divisors, the value {{math|2''n'' – ''σ''(''n'')}} is called the number's '''deficiency'''.  In terms of the aliquot sum {{math|''s''(''n'')}}, the deficiency is {{math|''n'' – ''s''(''n'')}}.

==Examples== 

The first few deficient numbers are
:1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... {{OEIS|id=A005100}}
As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 42, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.

==Properties==
Since the aliquot sums of prime numbers equal 1, all [[prime number]]s are deficient.{{sfnp|Prielipp|1970|loc=Theorem 1, pp. 693–694}} More generally, all odd numbers with one or two distinct prime factors are deficient.  It follows that there are infinitely many [[odd number|odd]] deficient numbers.  There are also an infinite number of [[even number|even]] deficient numbers as all [[Power of two|powers of two]] have the sum ({{math|1 + 2 + 4 + 8 + ... + 2{{sup|''x''-1}} {{=}} 2{{sup|''x''}} - 1}}).

More generally, all [[prime power]]s <math>p^k</math> are deficient, because their only proper divisors are <math>1, p, p^2, \dots, p^{k-1}</math> which sum to <math>\frac{p^k-1}{p-1}</math>, which is at most <math>p^k-1</math>.{{sfnp|Prielipp|1970|loc=Theorem 2, p. 694}}

All proper [[divisor]]s of deficient numbers are deficient.{{sfnp|Prielipp|1970|loc=Theorem 7, p. 695}}  Moreover, all proper divisors of [[perfect number]]s are deficient.{{sfnp|Prielipp|1970|loc=Theorem 3, p. 694}}

There exists at least one deficient number in the interval <math>[n, n + (\log n)^2]</math> for all sufficiently large ''n''.{{sfnp|Sándor|Mitrinović|Crstici|2006|p=108}}

==Related concepts==
{{Euler_diagram_numbers_with_many_divisors.svg}}
Closely related to deficient numbers are [[perfect number]]s with ''σ''(''n'')&nbsp;=&nbsp;2''n'', and [[abundant number]]s with ''σ''(''n'')&nbsp;>&nbsp;2''n''. 

[[Nicomachus]] was the first to subdivide numbers into deficient, perfect, or abundant, in his ''[[Introduction to Arithmetic]]'' (circa 100 CE). However, he applied this classification only to the [[even number]]s.{{sfnp|Dickson|1919|p=3}}

== See also ==

* [[Almost perfect number]]
* [[Amicable number]]
* [[Sociable number]]
* [[Superabundant number]]

==Notes==
{{reflist}}

==References==
*{{cite book|title=History of the Theory of Numbers, Vol. I: Divisibility and Primality|first=Leonard Eugene|last=Dickson|author-link=Leonard Eugene Dickson|publisher=Carnegie Institute of Washington|year=1919|url=https://archive.org/details/historyoftheoryo01dick}}
*{{cite journal | title=Perfect numbers, abundant numbers, and deficient numbers | year=1970 | journal=The Mathematics Teacher | volume=63 | number=8 | pages=692–696 | last=Prielipp|first= Robert W.| doi=10.5951/MT.63.8.0692 |jstor=27958492}}
* {{cite book | editor1-last=Sándor | editor1-first=József | editor2-last=Mitrinović | editor2-first=Dragoslav S. | editor3-last=Crstici |editor3-first=Borislav | title=Handbook of number theory I | location=Dordrecht | publisher=[[Springer-Verlag]] | year=2006 | isbn=1-4020-4215-9 | zbl=1151.11300 }}

== External links ==
* [http://primes.utm.edu/glossary/page.php?sort=DeficientNumber The Prime Glossary: Deficient number]
* {{MathWorld |urlname=DeficientNumber |title=Deficient Number}}
* {{PlanetMath |urlname=DeficientNumber |title=deficient number |id=7868}}

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

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