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Revision as of 08:22, 26 September 2017 by 158.169.40.5(talk)(→Examples: The number 32 did not correspond with the value for the sum of divisors, which is 11 in the example.)
Demonstration, with Cuisenaire rods, of the deficiency of the number 8
In number theory, a deficient or deficient number is a number n for which the sum of divisorsσ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is called the number's deficiency.
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 2 × 21 − 11 = 31.
Properties
Since the aliquot sums of prime numbers equal 1, all prime numbers are deficient.
An infinite number of both even and odd deficient numbers exist.
All odd numbers with one or two distinct prime factors are deficient.
There exists at least one deficient number in the interval for all sufficiently large n.[1]
Related concepts
Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n. The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100).
![{\displaystyle [n,n+(\log n)^{2}]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/30e4c49e65391a126244afffa7a3544a359cdfec)
