Arithmetic number
Integer where the average of its positive divisors is also an integer
title: "Arithmetic number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["divisor-function", "integer-sequences"] description: "Integer where the average of its positive divisors is also an integer" topic_path: "general/divisor-function" source: "https://en.wikipedia.org/wiki/Arithmetic_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Integer where the average of its positive divisors is also an integer ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/4/45/Arithmetic_number_Cuisenaire_rods_6.png" caption="Demonstration, with [[Cuisenaire rods]], of the arithmetic nature of the number 6"] ::
In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is :\frac{1+2+3+6}{4}=3, which is also an integer. However, 2 is not an arithmetic number because its only divisors are 1 and 2, and their average 3/2 is not an integer.
The first numbers in the sequence of arithmetic numbers are :1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, ... .
The arithmetic means of the divisors of arithmetic numbers are listed at .
Density
It is known that the natural density of such numbers is 1: indeed, the proportion of numbers less than X which are not arithmetic is asymptotically
:\exp\left( { -c \sqrt{\log\log X} } ,\right)
where c = 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 obeying the stronger condition that d(N)2 divides σ(N) is 1/2.
Notes
References
References
- Guy (2004) p.76
- (1981). "Analytic number theory, Proc. Conf., Temple Univ., 1980". [[Springer-Verlag]].
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