Arithmetic number

Short description: 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+64=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:[1] indeed, the proportion of numbers less than X which are not arithmetic is asymptotically[2]

exp⁡(−clog⁡log⁡X)

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 )2 divides σ(N ) is 1/2.[1][2]

## Notes

1. ↑ 1.0 1.1 Guy (2004) p.76
2. ↑ 2.0 2.1 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. https://www.math.dartmouth.edu/~carlp/PDF/31.pdf.

## References

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

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Classes of natural numbers

Powers and
related numbers|

* Achilles
* Power of 2
* Power of 10
* Square
* Cube
* Fourth power
* Fifth power
* Sixth power
* Seventh power
* Perfect power
* Powerful
* Prime power

Of the form
a × 2b ± 1|

* Cullen
* Double Mersenne
* Fermat
* Mersenne
* Proth
* Thabit
* Woodall

Other polynomial
numbers|

* Carol
* Hilbert
* Idoneal
* Kynea
* Leyland
* Loeschian
* Lucky numbers of Euler

Recursively defined
numbers|

* Fibonacci
* Jacobsthal
* Leonardo
* Lucas
* Padovan
* Pell
* Perrin

Possessing a
specific set
of other numbers|

* Knödel
* Riesel
* Sierpinski

Expressible via
specific sums|

* Nonhypotenuse
* Polite
* Practical
* Primary pseudoperfect
* Ulam
* Wolstenholme

Figurate numbers|

| 2-dimensional|

| centered|

* Centered triangular
* Centered square
* Centered pentagonal
* Centered hexagonal
* Centered heptagonal
* Centered octagonal
* Centered nonagonal
* Centered decagonal
* Star

|

non-centered|

* Triangular
* Square
* Square triangular
* Pentagonal
* Hexagonal
* Heptagonal
* Octagonal
* Nonagonal
* Decagonal
* Dodecagonal

3-dimensional|

| centered|

* Centered tetrahedral
* Centered cube
* Centered octahedral
* Centered dodecahedral
* Centered icosahedral

|

non-centered|

* Tetrahedral
* Cube
* Octahedral
* Dodecahedral
* Icosahedral
* Stella octangula

pyramidal|

* Square pyramidal
* Pentagonal pyramidal
* Hexagonal pyramidal
* Heptagonal pyramidal

4-dimensional|

| centered|

* Centered pentachoric
* Squared triangular

|

non-centered|

* Pentatope
* 4-cube

Base-dependent
numbers|

| Arithmetic functions and dynamics|

* Digit sum
* Digital root
* Multiplicative
* Dudeney
* Factorion
* Happy
* Kaprekar
* Kaprekar's constant
* Keith
* Lychrel
* Meertens
* Narcissistic
* Perfect digit-to-digit invariant
* Perfect digital invariant
* Persistence
* Additive
* Multiplicative
* Self
* Sum-product

|

P-adic numbers related|

* Automorphic

Divisor related|

* Equidigital
* Extravagant
* Frugal
* Harshad
* Polydivisible
* Smith

Digit-composition related|

* Palindromic
* Pandigital
* Repdigit
* Repunit
* Self-descriptive
* Smarandache–Wellin
* Strictly non-palindromic
* Undulating

Digit-permutation related|

* Cyclic
* Digit-reassembly
* Parasitic
* Primeval
* Transposable

Other|

* Friedman
* Vampire

Combinatorial
numbers|

* Bell
* Cake
* Catalan
* Dedekind
* Delannoy
* Euler
* Fuss–Catalan
* Lazy caterer's sequence
* Lobb
* Motzkin
* Narayana
* Ordered Bell
* Schröder
* Schröder–Hipparchus

Pseudoprimes|

* Carmichael number
* Catalan pseudoprime
* Elliptic pseudoprime
* Euler pseudoprime
* Euler–Jacobi pseudoprime
* Fermat pseudoprime
* Frobenius pseudoprime
* Lucas pseudoprime
* Somer–Lucas pseudoprime
* Strong pseudoprime

Arithmetic functions|

| By properties of σ(n)|

* Abundant
* Arithmetic
* Colossally abundant
* Descartes
* Hemiperfect
* Highly abundant
* Highly composite
* Hyperperfect
* Multiply perfect
* Perfect
* Practical number
* Primitive abundant
* Quasiperfect
* Refactorable
* Sublime
* Superabundant
* Superior highly composite
* Superperfect
* Untouchable

|

By properties of Ω(n)|

* Almost prime
* Semiprime

By properties of φ(n)|

* Highly cototient
* Highly totient
* Noncototient
* Nontotient
* Perfect totient
* Sparsely totient

By properties of s(n)|

* Almost perfect
* Amicable
* Betrothed
* Deficient
* Perfect
* Semiperfect
* Sociable

Other|

* Euclid
* Fortunate

Dividing a quotient|

* Wieferich
* Wall–Sun–Sun
* Wolstenholme prime
* Wilson

Other prime factor
or divisor related
numbers|

* Blum
* Erdős–Nicolas
* Erdős–Woods
* Friendly
* Giuga
* Harmonic divisor
* Lucas–Carmichael
* Pronic
* Regular
* Rough
* Smooth
* Sociable
* Sphenic
* Størmer
* Super-Poulet
* Zeisel

Binary numbers|

* Evil
* Odious
* Pernicious

Generated
via a sieve|

* Lucky
* Prime

Sorting related|

* Pancake number
* Sorting number

Natural language related|

* Aronson's sequence
* Ban

Graphemics related|

* Strobogrammatic

* Mathematics portal

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* e

Divisibility-based sets of integers

Overview|

* Integer factorization
* Divisor
* Unitary divisor
* Divisor function
* Prime factor
* Fundamental theorem of arithmetic
* Arithmetic number

Factorization forms|

* Prime
* Composite
* Semiprime
* Pronic
* Sphenic
* Square-free
* Powerful
* Perfect power
* Achilles
* Smooth
* Regular
* Rough
* Unusual

Constrained divisor sums|

* Perfect
* Almost perfect
* Quasiperfect
* Multiply perfect
* Hemiperfect
* Hyperperfect
* Superperfect
* Unitary perfect
* Semiperfect
* Practical
* Erdős–Nicolas

With many divisors|

* Abundant
* Primitive abundant
* Highly abundant
* Superabundant
* Colossally abundant
* Highly composite
* Superior highly composite
* Weird

Aliquot sequence-related|

* Untouchable
* Amicable
* Sociable
* Betrothed

Base-dependent|

* Equidigital
* Extravagant
* Frugal
* Harshad
* Polydivisible
* Smith

Other sets|

* Deficient
* Friendly
* Solitary
* Sublime
* Harmonic divisor

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