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{{TEX|done}}{{MSC|11A}}

An [[integer]] for which the [[arithmetic mean]] of its positive [[divisor]]s, is an integer. The first numbers in the [[sequence]] are 

$$1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, \ldots$$

which is {{OEIS|A003601}}.

It is known that the [[natural density]] of such numbers is 1 {{cite|Guy (2004) p.76}}. Indeed, the proportion of numbers less than $X$ which are not arithmetic is [[Asymptotic analysis|asymptotically]] {{cite|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.

==References==
* Bateman, Paul T.; Erdős, Paul; Pomerance, Carl; Straus, E.G. "The arithmetic mean of the divisors of an integer". In Knopp, M.I.. ''Analytic number theory, Proc. Conf., Temple Univ., 1980''. Lecture Notes in Mathematics '''899''' Springer-Verlag (1981) pp. 197–220. {{ZBL|0478.10027}}
* Guy, Richard K. ''Unsolved problems in number theory'' (3rd ed.). Springer-Verlag (2004).  {{ISBN|978-0-387-20860-2}} {{ZBL|1058.11001}}.  Section B2.