Arithmetic number

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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 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]

 \exp\left( { -c \sqrt{\log\log X} } \right)

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]


  1. ^ a b Guy (2004) p.76
  2. ^ 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.. Analytic number theory, Proc. Conf., Temple Univ., 1980. Lecture Notes in Mathematics 899. Springer-Verlag. pp. 197–220. Zbl 0478.10027.