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Difference between revisions of "Arithmetic number"

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An [[integer]] for which the [[arithmetic mean]] of its positive [[divisor]]s, is an integer. The first numbers in the [[sequence]] are  
 
An [[integer]] for which the [[arithmetic mean]] of its positive [[divisor]]s, is an integer. The first numbers in the [[sequence]] are  
  
$$1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, \ldots$$ <!--{{OEIS|id=A003601}}-->
+
$$1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, \ldots$$
 +
 
 +
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)}}
 
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)}}

Latest revision as of 09:52, 12 October 2023

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

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

which is OEIS sequence A003601.

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.
How to Cite This Entry:
Arithmetic number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_number&oldid=53993