In number theory, an arithmetic number is an integer for which the average of its positivedivisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is
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which is also an integer. Code pandorum reborn free download. 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
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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 divisorsd(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.[1][2]
Notes[edit]
^ abGuy (2004) p.76
^ abBateman, 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. 899. Springer-Verlag. pp. 197–220. Zbl0478.10027.
References[edit]
Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. B2. ISBN978-0-387-20860-2. Zbl1058.11001.
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Arithmetic_number&oldid=826635033'
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