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Difference between revisions of "Arithmetic number"

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(Start article: Arithmetic number)
 
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An [[integer]] for which the [[arithmetic mean]] of its [[positive number|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 {{OEIS|id=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)}}
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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  
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$$1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, \ldots$$
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which is {{OEIS|A003601}}.
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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)
 
  \exp\left( { -c \sqrt{\log\log X} } \right)
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where $c = 2\sqrt{\log 2} + o(1)$.
 
where $c = 2\sqrt{\log 2} + o(1)$.
  
A number $N$ is arithmetic if the [[number of divisors]] $d(N)$ divides the [[Sum-of-divisors function|sum of divisors]] $\sigma(N)$. The natural density of integers $N$ for which $d(N)^2$ divides  $\sigma(N)$ is 1/2.
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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==
 
==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}}
 
* 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.
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* Guy, Richard K. ''Unsolved problems in number theory'' (3rd ed.). Springer-Verlag (2004).  {{ISBN|978-0-387-20860-2}} {{ZBL|1058.11001}}.  Section B2.

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=34855