{{short description|Sum of all proper divisors of a natural number}}
In number theory, the '''aliquot sum''' {{math|''s''(''n'')}} of a positive integer {{mvar|n}} is the sum of all proper divisors of {{mvar|n}}, that is, all divisors of {{mvar|n}} other than {{mvar|n}} itself. That is, <math display=block>s(n)=\sum_{{d|n,} \atop {d\ne n}} d \, .</math>
It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
==Examples== For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are {{nowrap|1, 2, 3, 4}}, and 6, so the aliquot sum of 12 is 16 i.e. ({{nowrap|1 + 2 + 3 + 4 + 6}}).
The values of {{math|''s''(''n'')}} for {{nowrap|1={{mvar|n}} = 1, 2, 3, ...}} are:
:0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ... {{OEIS|A001065}}
==Characterization of classes of numbers==
The aliquot sum function can be used to characterize several notable classes of numbers: *1 is the only number whose aliquot sum is 0. *A number is prime if and only if its aliquot sum is 1.{{r|pp}} *The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively.{{r|pp}} The quasiperfect numbers (if such numbers exist) are the numbers {{mvar|n}} whose aliquot sums equal {{math|''n'' + 1}}. The almost perfect numbers (which include the powers of 2, being the only known such numbers so far) are the numbers {{mvar|n}} whose aliquot sums equal {{math|''n'' − 1}}. *The untouchable numbers are the numbers that are not the aliquot sum of any other number. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable.{{r|pp|s}} Paul Erdős proved that their number is infinite.{{r|e}} The conjecture that 5 is the only odd untouchable number remains unproven, but would follow from a form of Goldbach's conjecture together with the observation that, for a semiprime number {{mvar|pq}}, the aliquot sum is {{math|''p'' + ''q'' + 1}}.{{r|pp}}
The mathematicians {{harvtxt|Pollack|Pomerance|2016}} noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
==Iteration== {{main|Aliquot sequence}} Iterating the aliquot sum function produces the aliquot sequence {{math|''n'', ''s''(''n''), ''s''(''s''(''n'')), …}} of a nonnegative integer {{mvar|n}} (in this sequence, we define {{math|1=''s''(0) = 0}}).
Sociable numbers are numbers whose aliquot sequence is a periodic sequence. Amicable numbers are sociable numbers whose aliquot sequence has period 2.
It remains unknown whether these sequences always end with a prime number, a perfect number, or a periodic sequence of sociable numbers.<ref>{{MathWorld | urlname=CatalansAliquotSequenceConjecture | title=Catalan's Aliquot Sequence Conjecture}}</ref>
== See also == * Sum of positive divisors function, the sum of the ({{mvar|x}}th powers of the) positive divisors of a number * William of Auberive, medieval numerologist interested in aliquot sums
==References== <references>
<ref name=e>{{citation | last = Erdős | first = P. | authorlink = Paul Erdős | journal = Elemente der Mathematik | mr = 0337733 | pages = 83–86 | title = Über die Zahlen der Form <math>\sigma(n)-n</math> und <math>n-\phi(n)</math> | url = https://users.renyi.hu/~p_erdos/1973-27.pdf | volume = 28 | year = 1973}}</ref>
<ref name=pp>{{citation | last1 = Pollack | first1 = Paul | last2 = Pomerance | first2 = Carl | author2-link = Carl Pomerance | doi = 10.1090/btran/10 | journal = Transactions of the American Mathematical Society | mr = 3481968 | pages = 1–26 | series = Series B | title = Some problems of Erdős on the sum-of-divisors function | volume = 3 | year = 2016| doi-access = free }}</ref>
<ref name=s>{{citation | last = Sesiano | first = J. | issue = 3 | journal = Archive for History of Exact Sciences | jstor = 41133889 | mr = 1107382 | pages = 235–238 | title = Two problems of number theory in Islamic times | volume = 41 | year = 1991 | doi = 10.1007/BF00348408| s2cid = 115235810 }}</ref>
</references>
==External links== *{{MathWorld|title=Restricted Divisor Function|id=RestrictedDivisorFunction}}
Category:Arithmetic dynamics Category:Arithmetic functions Category:Divisor function Category:Perfect numbers