# Aliquot sum > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Aliquot_sum > Markdown URL: https://mediated.wiki/source/Aliquot_sum.md > Source: https://en.wikipedia.org/wiki/Aliquot_sum > Source revision: 1328749187 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{short description|Sum of all proper divisors of a natural number}} In [number theory](/source/number_theory), the '''aliquot sum''' {{math|''s''(''n'')}} of a [positive integer](/source/positive_integer) {{mvar|n}} is the sum of all [proper divisors](/source/proper_divisors) of {{mvar|n}}, that is, all divisors of {{mvar|n}} other than {{mvar|n}} itself. That is, s(n)=\sum_{{d|n,} \atop {d\ne n}} d \, . It can be used to characterize the [prime number](/source/prime_number)s, [perfect number](/source/perfect_number)s, [sociable number](/source/sociable_number)s, [deficient number](/source/deficient_number)s, [abundant number](/source/abundant_number)s, and [untouchable number](/source/untouchable_number)s, and to define the [aliquot sequence](/source/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](/source/prime_number) if and only if its aliquot sum is 1.{{r|pp}} *The aliquot sums of [perfect](/source/Perfect_number), [deficient](/source/Deficient_number), and [abundant](/source/Abundant_number) numbers are equal to, less than, and greater than the number itself respectively.{{r|pp}} The [quasiperfect number](/source/quasiperfect_number)s (if such numbers exist) are the numbers {{mvar|n}} whose aliquot sums equal {{math|''n'' + 1}}. The [almost perfect number](/source/almost_perfect_number)s (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 number](/source/untouchable_number)s are the numbers that are not the aliquot sum of any other number. Their study goes back at least to [Abu Mansur al-Baghdadi](/source/Abu_Mansur_al-Baghdadi) (circa 1000 AD), who observed that both 2 and 5 are untouchable.{{r|pp|s}} [Paul Erdős](/source/Paul_Erd%C5%91s) 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](/source/Goldbach's_conjecture) together with the observation that, for a [semiprime number](/source/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](/source/Iterated_function) the aliquot sum function produces the [aliquot sequence](/source/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](/source/Sociable_number) are numbers whose aliquot sequence is a [periodic sequence](/source/periodic_sequence). [Amicable numbers](/source/Amicable_numbers) are sociable numbers whose aliquot sequence has period 2. It remains unknown whether these sequences always end with a [prime number](/source/prime_number), a [perfect number](/source/perfect_number), or a periodic sequence of sociable numbers.{{MathWorld | urlname=CatalansAliquotSequenceConjecture | title=Catalan's Aliquot Sequence Conjecture}} == See also == * [Sum of positive divisors function](/source/Divisor_function), the sum of the ({{mvar|x}}th powers of the) positive divisors of a number * [William of Auberive](/source/William_of_Auberive), medieval numerologist interested in aliquot sums ==References== {{citation | last = Erdős | first = P. | authorlink = Paul Erdős | journal = [Elemente der Mathematik](/source/Elemente_der_Mathematik) | mr = 0337733 | pages = 83–86 | title = Über die Zahlen der Form \sigma(n)-n und n-\phi(n) | url = https://users.renyi.hu/~p_erdos/1973-27.pdf | volume = 28 | year = 1973}} {{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 }} {{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 }} ==External links== *{{MathWorld|title=Restricted Divisor Function|id=RestrictedDivisorFunction}} Category:Arithmetic dynamics Category:Arithmetic functions Category:Divisor function Category:Perfect numbers --- Adapted from the Wikipedia article [Aliquot sum](https://en.wikipedia.org/wiki/Aliquot_sum) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Aliquot_sum?action=history)). 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