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In number theory, a deficient number or defective number is a positive integer $n$ for which the sum of divisors of $n$ is less than $2 n$ . Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than $n$ . For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.

Denoting by $σ (n)$ the sum of divisors, the value $2 n - σ (n)$ is called the number's deficiency. In terms of the aliquot sum $s (n)$ , the deficiency is $n - s (n)$ .

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, ... (sequence A005100 in the OEIS)

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 numbers are deficient.^[1] More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely many odd deficient numbers. There are also an infinite number of even deficient numbers as all powers of two have the sum ( $1 + 2 + 4 + 8 + ... + 2 x -1 = 2 x - 1$ ).

More generally, all prime powers $p^{k}$ are deficient, because their only proper divisors are $1,p,p^{2},\dots ,p^{k-1}$ which sum to ${\frac {p^{k}-1}{p-1}}$ , which is at most $p^{k}-1$ .^[2]

All proper divisors of deficient numbers are deficient.^[3] Moreover, all proper divisors of perfect numbers are deficient.^[4]

There exists at least one deficient number in the interval $[n,n+(\log n)^{2}]$ for all sufficiently large n.^[5]

Related concepts

Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n.

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 numbers.^[6]

Notes

^ Prielipp (1970), Theorem 1, pp. 693–694.
^ Prielipp (1970), Theorem 2, p. 694.
^ Prielipp (1970), Theorem 7, p. 695.
^ Prielipp (1970), Theorem 3, p. 694.
^ Sándor, Mitrinović & Crstici (2006), p. 108.
^ Dickson (1919), p. 3.

References

Dickson, Leonard Eugene (1919). History of the Theory of Numbers, Vol. I: Divisibility and Primality. Carnegie Institute of Washington.
Prielipp, Robert W. (1970). "Perfect numbers, abundant numbers, and deficient numbers". The Mathematics Teacher. 63 (8): 692–696. doi:10.5951/MT.63.8.0692. JSTOR 27958492.
Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.

External links

[FOOTNOTEPrielipp1970Theorem_1,_pp._693–694-1] Prielipp (1970), Theorem 1, pp. 693–694.

[FOOTNOTEPrielipp1970Theorem_2,_p._694-2] Prielipp (1970), Theorem 2, p. 694.

[FOOTNOTEPrielipp1970Theorem_7,_p._695-3] Prielipp (1970), Theorem 7, p. 695.

[FOOTNOTEPrielipp1970Theorem_3,_p._694-4] Prielipp (1970), Theorem 3, p. 694.

[FOOTNOTESándorMitrinovićCrstici2006108-5] Sándor, Mitrinović & Crstici (2006), p. 108.

[FOOTNOTEDickson19193-6] Dickson (1919), p. 3.

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@@ Line 1: / Line 1: @@
+{{short description|Number whose divisor sum is less than itself}}
-[[File:Deficient number Cuisenaire rods 8.png|thumb|Demonstration, with [[Cuisenaire rod]]s, of the deficiency of the number 8]]
+[[File:Deficient number Cuisenaire rods 8.png|thumb|Demonstration, with [[Cuisenaire rods]], of the deficiency of the number 8]]
-In [[number theory]], a '''deficient''' or '''deficient number''' is a number ''n'' for which the [[Divisor function#Definition|sum of divisors]] ''&sigma;''(''n'')<2''n'', or, equivalently, the sum of proper divisors (or [[aliquot sum]]) ''s''(''n'')<''n''. The value 2''n''&nbsp;&minus;&nbsp;''σ''(''n'') (or ''n''&nbsp;&minus;&nbsp;''s''(''n'')) is called the number's '''deficiency'''.
+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:
+The first few deficient numbers are
-:[[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]], [[29 (number)|29]], 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... {{OEIS|id=A005100}}
+: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 proper divisors are 1, 3 and 7, and their sum is 11. Because 11 is less than 21, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.
+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}}).
-*Since the aliquot sums of prime numbers equal 1, all [[prime number]]s are deficient.
-*An infinite number of both [[even and odd numbers|even and odd]] deficient numbers exist.
+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 odd numbers with one or two distinct prime factors are deficient.
-*All proper [[divisor]]s of deficient or [[perfect number]]s are deficient.
+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''.<ref name=HBI108>Sándor et al (2006) p.108</ref>
+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}}
-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''. The [[natural number]]s were first classified as either deficient, perfect or abundant by [[Nicomachus]] in his ''Introductio Arithmetica'' (circa 100 CE).
 == See also ==
@@ Line 25: / Line 32: @@
 * [[Amicable number]]
 * [[Sociable number]]
+* [[Superabundant number]]
-==References==
+==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 }}
@@ Line 38: / Line 50: @@
 {{Classes of natural numbers}}
+[[Category:Arithmetic dynamics]]
 [[Category:Divisor function]]
 [[Category:Integer sequences]]

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect