Aliquot sum#Definition

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}}

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.

{{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.{{MathWorld | urlname=CatalansAliquotSequenceConjecture | title=Catalan's Aliquot Sequence Conjecture}}

• 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

{{reflist|refs=

{{citation

| last = Erdős | first = P. | authorlink = Paul Erdős

| journal = Elemente der Mathematik

| mr = 0337733

| pages = 83–86

| title = Über die Zahlen der Form $\backslash sigma(n)-n$ und $n-\backslash phi(n)$

| 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

}}

}}

• {{MathWorld|title=Restricted Divisor Function|id=RestrictedDivisorFunction}}

Category:Arithmetic dynamics

Category:Arithmetic functions

Category:Divisor function

Category:Perfect numbers