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deficient numbers
A number    is said to be deficient if  , i.e., if the sum of the proper divisors of    is smaller than  .

For example, 7 is deficient, since its only proper divisor is 1, which is smaller than 7. It is easy to see that all the primes, their powers and all the semiprimes apart 6, are deficient.

Moreover, if a number is perfect or deficient, then all its proper divisors are deficient.

J.Sándor has proved that for    sufficiently large there is always a deficient number between    and  .

About 3/4 of the integers are deficient. The first deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22 more terms

Below, the spiral pattern of deficient numbers up to 2500. See the page on prime numbers for an explanation and links to similar pictures.

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

a-pointer ABA aban Achilles alt.fact. alternating amenable amicable anti-perfect apocalyptic arithmetic 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 dig.balanced double fact. Duffinian eban economical emirp emirpimes enlightened equidigital eRAP esthetic Eulerian evil fibodiv Fibonacci Friedman frugal gapful Gilda good prime happy Harshad heptagonal hex hexagonal hoax Hogben Honaker house hungry hyperperfect iban iccanobiF idoneal impolite 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 prime primeval pronic Proth rare repdigit repfigit repunit Rhonda Ruth-Aaron Sastry self self-describing semiprime sliding Smith Sophie Germain sphenic square star straight-line strobogrammatic strong prime subfactorial super Niven super-d tau taxicab tetrahedral tetranacci triangular tribonacci trimorphic truncatable prime twin uban Ulam undulating unprimeable untouchable upside-down vampire wasteful weak prime weakly prime Wieferich Woodall Zeisel Zuckerman zygodrome