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| In [[number theory]], a '''deficient number''' or '''defective number''' is a number ''n'' for which the [[Divisor_function#Definition|sum of divisors]] ''σ''(''n'')<2''n'', or, equivalently, the sum of proper divisors (or [[aliquot sum]]) ''s''(''n'')<''n''. The value 2''n'' − ''σ''(''n'') (or ''n'' − ''s''(''n'')) is called the number's '''deficiency'''.
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| ==Examples==
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| The first few deficient numbers are:
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| :[[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[5 (number)|5]], [[7 (number)|7]], [[8 (number)|8]], [[9 (number)|9]], [[10 (number)|10]], [[11 (number)|11]], [[13 (number)|13]], [[14 (number)|14]], [[15 (number)|15]], [[16 (number)|16]], [[17 (number)|17]], [[19 (number)|19]], [[21 (number)|21]], [[22 (number)|22]], [[23 (number)|23]], [[25 (number)|25]], [[26 (number)|26]], [[27 (number)|27]], … {{OEIS|id=A005100}}
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| 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 2 × 21, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.
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| ==Properties==
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| *An infinite number of both [[even and odd numbers|even and odd]] deficient numbers exist
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| *All odd numbers with one or two distinct prime factors are deficient
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| *All proper [[divisor]]s of deficient or [[perfect number]]s are deficient.
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| *There exists at least one deficient number in the interval <math>[n, n + (\log n)^2]</math> for all sufficiently large ''n''.<ref name=HBI108>Sándor et al (2006) p.108</ref>
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| ==Related concepts== | |
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| Closely related to deficient numbers are [[perfect number]]s with ''σ''(''n'') = 2''n'', and [[abundant number]]s with ''σ''(''n'') > 2''n''. The [[natural number]]s were first classified as either deficient, perfect or abundant by [[Nicomachus]] in his ''Introductio Arithmetica'' (circa 100).
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| == See also ==
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| * [[Almost perfect number]]
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| * [[Amicable number]]
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| * [[Sociable number]]
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| ==References==
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| {{reflist}}
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| * {{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 }}
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| == External links ==
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| * [http://primes.utm.edu/glossary/page.php?sort=DeficientNumber The Prime Glossary: Deficient number]
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| * {{MathWorld |urlname=DeficientNumber |title=Deficient Number}}
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| * {{PlanetMath |urlname=DeficientNumber |title=deficient number |id=7868}}
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| {{Divisor classes}}
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| {{Classes of natural numbers}}
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| [[Category:Divisor function]]
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| [[Category:Integer sequences]] | |
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