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arithmetic numbers
A number    is called arithmetic if the arithmetic mean of its divisors is an integer number.

For example, 42 is arithmetic because it has 8 divisors (1, 2, 3, 6, 7, 14, 21, 42) whose average is 96/8 = 12.

All the odd prime numbers are arithmetic. It is possible to find quite long runs of consecutive arithmetic numbers. For example, one of length 105 starts at 3033935561.

Antonio M. Oller-MarcĂ©n has proved various properties of arithmetic numbers, for example, that if    is an odd prime then    is arithmetic.

Below, the spiral pattern of arithmetic numbers up to  . See the page on prime numbers for an explanation and links to similar pictures.

The smallest 3 × 3 magic square made of consecutive arithmetic numbers is

 47 39 46 43 44 45 42 49 41

The first 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, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70 more terms

Arithmetic numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer ABA aban abundant Achilles admirable alt.fact. alternating amenable amicable anti-perfect apocalyptic astonishing automorphic balanced p. Bell bemirp betrothed binomial brilliant c.decagonal c.heptagonal c.nonagonal c.octagonal c.pentagonal c.square c.triangular cake Canada Carmichael Carol Catalan Chen congruent constructible cube Cullen Cunningham Curzon cyclic D-number d-powerful de Polignac decagonal deceptive deficient dig.balanced double fact. droll Duffinian eban economical emirp emirpimes enlightened equidigital eRAP esthetic Eulerian evil fibodiv Fibonacci Friedman frugal gapful Gilda Giuga good prime happy harmonic Harshad heptagonal hex hexagonal highly composite hoax Hogben Honaker house hungry hyperperfect iban iccanobiF idoneal inconsummate insolite interprime Jacobsthal Jordan-Polya junction Kaprekar katadrome Kynea Lehmer Leyland lonely Lucas lucky Lynch-Bell m-pointer magic magnanimous metadrome modest Moran Motzkin narcissistic nialpdrome nonagonal nude O'Halloran oban octagonal odious Ormiston palindromic palprime pancake panconsummate pandigital partition pentagonal pernicious Perrin Pierpont plaindrome Poulet power powerful practical prim.abundant prime primeval primorial pronic Proth pseudoperfect rare repdigit repfigit repunit Rhonda Ruth-Aaron Saint-Exupery Sastry self self-describing semiprime sliding Smith Sophie Germain sphenic square star straight-line strobogrammatic strong prime subfactorial super Niven super-d superabundant tau taxicab tcefrep tetrahedral tetranacci triangular tribonacci trimorphic truncatable prime twin uban Ulam undulating unprimeable untouchable upside-down vampire wasteful weak prime weakly prime weird Wieferich Woodall Zeisel Zuckerman Zumkeller zygodrome