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
| 1+2+3+6 | |
| 4 |
=3,
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 .
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]
\exp\left({-c\sqrt{loglogX}}\right)
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.[1] [2]
. Richard K. Guy . Unsolved problems in number theory . . 3rd . 2004 . 978-0-387-20860-2 . 1058.11001 . B2.