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
 * $$\frac{1+2+3+6}{4}=3,$$

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, ....

The arithmetic means of the divisors of arithmetic numbers are listed at.

Density
It is known that the natural density of such numbers is 1: indeed, the proportion of numbers less than X which are not arithmetic is asymptotically


 * $$\exp\left( { -c \sqrt{\log\log X} } \,\right)$$

where c = 2$\sqrt{log 2}$ + o(1).

A number N is arithmetic if the number of divisors d(N&hairsp;) divides the sum of divisors σ(N&hairsp;). It is known that the density of integers N obeying the stronger condition that d(N&hairsp;)2 divides σ(N&hairsp;) is 1/2.