Arithmetic number
In mathematics, an arithmetic number is an integer for which the arithmetic mean of its positive divisors is an integer. It is known that the natural density of such numbers is 1:[1] indeed, the proportion of numbers less than X which are not arithmetic is asymptotically[2]
where c = 2 √ log 2 + o(1).
References
- ^ Guy (2004) p.76
- ^ Bateman, Paul T.; Erdős, Paul; Pomerance, Carl; Straus, E.G. (1981). "The arithmetic mean of the divisors of an integer". In Knopp, M.I. (ed.). Analytic number theory, Proc. Conf., Temple Univ., 1980. Lecture Notes in Mathematics. Vol. 899. Springer-Verlag. pp. 197–220. Zbl 0478.10027.
- Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B2. ISBN 978-0-387-20860-2. Zbl 1058.11001.
Divisibility-based sets of integers | ||
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| Overview |  | |
| Factorization forms | ||
| Constrained divisor sums | ||
| With many divisors | ||
| Aliquot sequence-related | ||
| Base-dependent | ||
| Other sets | ||
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