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
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
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
where c = 2√ + 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.