In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, the aliquot sum ''s''(''n'') of a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
''n'' is the sum of all proper
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of ''n'', that is, all divisors of ''n'' other than ''n'' itself.
That is,
:
It can be used to characterize the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s,
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
...
s, "
sociable numbers",
deficient numbers,
abundant number
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. Th ...
s, 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 1, 2, 3, 4, and 6, so the aliquot sum of 12 is 16 i.e. (1 + 2 + 3 + 4 + 6).
The values of ''s''(''n'') for ''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, ...
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
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
if and only if its aliquot sum is 1.
*The aliquot sums of
perfect,
deficient, and
abundant numbers are equal to, less than, and greater than the number itself respectively. The
quasiperfect numbers (if such numbers exist) are the numbers ''n'' whose aliquot sums equal ''n'' + 1. The
almost perfect numbers (which include the powers of 2, being the only known such numbers so far) are the numbers ''n'' whose aliquot sums equal ''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.
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
proved that their number is infinite. The conjecture that 5 is the only odd untouchable number remains unproven, but would follow from a form of
Goldbach's conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
together with the observation that, for a
semiprime number ''pq'', the aliquot sum is ''p'' + ''q'' + 1.
The mathematicians noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
Iteration
Iterating the aliquot sum function produces the
aliquot sequence ''n'', ''s''(''n''), ''s''(''s''(''n'')), ... of a nonnegative integer ''n'' (in this sequence, we define ''s''(0) = 0). It remains unknown whether these sequences always end with a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
, a
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
...
, or a
periodic sequence of
sociable numbers.
See also
*
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...
: The sum of the (''x''th powers of the) positive divisors of a number
*
William of Auberive, medieval numerologist interested in aliquot sums
References
External links
*{{MathWorld, title=Restricted Divisor Function, id=RestrictedDivisorFunction
Arithmetic dynamics
Arithmetic functions
Divisor function
Perfect numbers