Arithmetic number
In number theory, an arithmetic number is 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 (sequence A003601 in the OEIS). 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).
A number N is arithmetic if the number of divisors d(N) divides the sum of divisors σ(N). It is known that the density of integers N for which d(N)2 divides σ(N) is 1/2.[1][2]
Notes
- ^ a b Guy (2004) p.76
- ^ a b 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 (PDF). Lecture Notes in Mathematics. Vol. 899. Springer-Verlag. pp. 197–220. Zbl 0478.10027.
References
- 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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