Aliquot sum

In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is,

${\displaystyle s(n)=\sum \nolimits _{d|n,\ d\neq n}d.}$

It can be used to characterize the prime numbers, perfect numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.

Examples[]

For example, the proper divisors of 15 (that is, the positive divisors of 15 that are not equal to 15) are 1, 3 and 5, so the aliquot sum of 15 is 9 i.e. (1 + 3 + 5).

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, ... (sequence A001065 in the OEIS)

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.^[1]
• The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively.^[1] 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.^[1][2] Paul Erdős proved that their number is infinite.^[3] 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 pq, the aliquot sum is p + q + 1.^[1]

The mathematicians Pollack & Pomerance (2016) 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 converge (the limit of the sequence must be 0 or a perfect number), or whether they can diverge (i.e. the limit of the sequence does not exist).^[1]

See also[]

•  : The sum of the (xth powers of the) positive divisors of a number
• William of Auberive, medieval numerologist interested in aliquot sums

References[]

External links[]

• Weisstein, Eric W. "Restricted Divisor Function". MathWorld.