Английская Википедия:Arithmetic number
In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is
- <math>\frac{1+2+3+6}{4}=3,</math>
which is also an integer. However, 2 is not an arithmetic number because its only divisors are 1 and 2, and their average 3/2 is not an integer.
The first numbers in the sequence of arithmetic numbers are
- 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, ... Шаблон:OEIS.
Density
It is known that the natural density of such numbers is 1:[1] indeed, the proportion of numbers less than X which are not arithmetic is asymptotically[2]
- <math>\exp\left( { -c \sqrt{\log\log X} } \,\right)</math>
where c = 2Шаблон:Radic + o(1).
A number N is arithmetic if the number of divisors d(N ) divides the sum of divisors σ(N ). It is known that the density of integers N obeying the stronger condition that d(N )2 divides σ(N ) is 1/2.[1][2]
Notes
References
Шаблон:Classes of natural numbers Шаблон:Divisor classes
- ↑ 1,0 1,1 Guy (2004) p.76
- ↑ 2,0 2,1 Шаблон:Cite book
