In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, the aliquot sum of a
positive integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
is the sum of all
proper divisors of , that is, all divisors of other than 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 proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...
s,
sociable numbers,
deficient numbers,
abundant number
In number theory, an abundant number or excessive number is a positive integer 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 ...
s, and
untouchable numbers, and to define the
aliquot sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
Def ...
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 , and 6, so the aliquot sum of 12 is 16 i.e. ().
The values of for 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 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 whose aliquot sums equal . The
almost perfect numbers (which include the powers of 2, being the only known such numbers so far) are the numbers whose aliquot sums equal .
*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 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 list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the ...
together with the observation that, for a
semiprime number , the aliquot sum is .
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
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
Def ...
of a nonnegative integer (in this sequence, we define ).
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 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 proper divisors, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfec ...
, or a periodic sequence of sociable numbers.
See also
*
Sum of positive divisors function, the sum of the (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