Misplaced Pages

Arithmetic number

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Integer where the average of its positive divisors is also an integer
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

1 + 2 + 3 + 6 4 = 3 , {\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 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, ... (sequence A003601 in the OEIS).

The arithmetic means of the divisors of arithmetic numbers are listed at A102187.

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 ( c log log X ) {\displaystyle \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 obeying the stronger condition that d(N ) divides σ(N ) is 1/2.

Notes

  1. ^ Guy (2004) p.76
  2. ^ 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

Classes of natural numbers
Powers and related numbers
Of the form a × 2 ± 1
Other polynomial numbers
Recursively defined numbers
Possessing a specific set of other numbers
Expressible via specific sums
Figurate numbers
2-dimensional
centered
non-centered
3-dimensional
centered
non-centered
pyramidal
4-dimensional
non-centered
Combinatorial numbers
Primes
Pseudoprimes
Arithmetic functions and dynamics
Divisor functions
Prime omega functions
Euler's totient function
Aliquot sequences
Primorial
Other prime factor or divisor related numbers
Numeral system-dependent numbers
Arithmetic functions
and dynamics
Digit sum
Digit product
Coding-related
Other
P-adic numbers-related
Digit-composition related
Digit-permutation related
Divisor-related
Other
Binary numbers
Generated via a sieve
Sorting related
Natural language related
Graphemics related
Divisibility-based sets of integers
Overview Divisibility of 60
Factorization forms
Constrained divisor sums
With many divisors
Aliquot sequence-related
Base-dependent
Other sets
Categories: