Английская Википедия:Aliquot sum
In number theory, the aliquot sum Шаблон:Math of a positive integer Шаблон:Mvar is the sum of all proper divisors of Шаблон:Mvar, that is, all divisors of Шаблон:Mvar other than Шаблон:Mvar itself. That is, <math display=block>s(n)=\sum_Шаблон:D d \, .</math>
It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, 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 Шаблон:Nowrap, and 6, so the aliquot sum of 12 is 16 i.e. (Шаблон:Nowrap).
The values of Шаблон:Math for Шаблон:Nowrap 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, ... Шаблон: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.Шаблон:R
- The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively.Шаблон:R The quasiperfect numbers (if such numbers exist) are the numbers Шаблон:Mvar whose aliquot sums equal Шаблон:Math. The almost perfect numbers (which include the powers of 2, being the only known such numbers so far) are the numbers Шаблон:Mvar whose aliquot sums equal Шаблон:Math.
- 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.Шаблон:R Paul Erdős proved that their number is infinite.Шаблон:R 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 Шаблон:Mvar, the aliquot sum is Шаблон:Math.Шаблон:R
The mathematicians Шаблон:Harvtxt noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
Iteration
Шаблон:Main Iterating the aliquot sum function produces the aliquot sequence Шаблон:Math of a nonnegative integer Шаблон:Mvar (in this sequence, we define Шаблон:Math).
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.[1]
See also
- Sum of positive divisors function, the sum of the (Шаблон:Mvarth powers of the) positive divisors of a number
- William of Auberive, medieval numerologist interested in aliquot sums
References
External links
