# Arithmetic number

An integer for which the arithmetic mean of its positive divisors, is an integer. The first numbers in the sequence are 1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20 . It is known that the natural density of such numbers is 1:[Guy (2004) p.76] indeed, the proportion of numbers less than $X$ which are not arithmetic is asymptotically[Bateman et al (1981)] $$\exp\left( { -c \sqrt{\log\log X} } \right)$$ where $c = 2\sqrt{\log 2} + o(1)$.
A number $N$ is arithmetic if the number of divisors $\tau(N)$ divides the sum of divisors $\sigma(N)$. The natural density of integers $N$ for which $d(N)^2$ divides $\sigma(N)$ is 1/2.