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Deficient number
Number that is more than the sum of its proper divisors
Number that is more than the sum of its proper divisors
In number theory, a deficient number or defective number is a positive integer n for which the sum of divisors of n is less than 2n. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than n. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient.
Denoting by σ(n) the sum of divisors, the value 2n − σ(n) is called the number's deficiency. In terms of the aliquot sum s(n), the deficiency is n − s(n).
Examples
The first few deficient numbers are :1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... As an example, consider the number 21. Its proper divisors are 1, 3 and 7, and their sum is 11. Because 11 is less than 21, the number 21 is deficient. Its deficiency is 21 − 11 = 10.
Properties
Since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely many odd deficient numbers. There are also an infinite number of even deficient numbers as all powers of two have the sum (). The infinite family of numbers of form 2n − 1 × p**m where m 0 and p is a prime 2n − 1 are also deficient.
More generally, all prime powers p^k are deficient, because their only proper divisors are 1, p, p^2, \dots, p^{k-1} which sum to \frac{p^k-1}{p-1}, which is at most p^k-1.
All proper divisors of deficient numbers are deficient. Moreover, all proper divisors of perfect numbers are deficient.
There exists at least one deficient number in the interval [n, n + (\log n)^2] for all sufficiently large n.
Notes
References
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