Difference between revisions of "Arithmetic number"
From Encyclopedia of Mathematics
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| − | An [[integer]] for which the [[arithmetic mean]] of its | + | 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 <!--{{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)}} |
$$ | $$ | ||
\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]] $ | + | 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. | * Guy, Richard K. ''Unsolved problems in number theory'' (3rd ed.). Springer-Verlag (2004). ISBN 978-0-387-20860-2 {{ZBL|1058.11001}}. Section B2. | ||
Revision as of 18:09, 28 December 2014
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 . 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=35916
Arithmetic number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_number&oldid=35916