Innumber theory, anarithmetic number is aninteger for which theaverage of itspositivedivisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is

${\displaystyle \frac{1+2+3+6}{4}=3,}$

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 thesequence 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, ... (sequenceA003601 in theOEIS).

Thearithmetic means of the divisors of arithmetic numbers are listed atA102187.

It is known that thenatural density of such numbers is 1:^[1] indeed, the proportion of numbers less thanX which are not arithmetic isasymptotically^[2]

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

wherec = 2√log 2 + o(1).

A numberN is arithmetic if thenumber of divisorsd(N ) divides thesum of divisors σ(N ). It is known that thedensity of integersN obeying the stronger condition thatd(N )² divides σ(N ) is 1/2.^[1][2]

• Guy, Richard K. (2004).Unsolved problems in number theory (3rd ed.).Springer-Verlag. B2.ISBN 978-0-387-20860-2.Zbl 1058.11001.

• Divisor function
• Integer sequences

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