Difference between revisions of "Arithmetic number"
From Encyclopedia of Mathematics
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| − | An [[integer]] for which the [[arithmetic mean]] of its positive [[divisor]]s, | + | 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]] $\tau(N)$ divides the [[sum of divisors]] $\sigma(N)$. The | + | 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 07:13, 16 March 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
$$1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20 .$$
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=52632
Arithmetic number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetic_number&oldid=52632