Arithmetic number

2020 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

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

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

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.

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.