Deficient number

File:Deficient number Cuisenaire rods 8.png

Demonstration, with Cuisenaire rods, of the deficiency of the number 8

In number theory, a deficient number or defective number is a positive integer Template:Mvar for which the sum of divisors of Template:Mvar is less than Template:Math. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than Template:Mvar. For example, the proper divisors of 8 are Template:Nowrap, and their sum is less than 8, so 8 is deficient.

Denoting by Template:Math the sum of divisors, the value Template:Math is called the number's deficiency. In terms of the aliquot sum Template:Math, the deficiency is Template:Math.

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, ... Template:OEIS

As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 42, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.

Properties

Since the aliquot sums of prime numbers equal 1, all prime numbers are deficient.Template:Sfnp 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 (Template:Math).

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 $p^k-1$.Template:Sfnp

All proper divisors of deficient numbers are deficient.Template:Sfnp Moreover, all proper divisors of perfect numbers are deficient.Template:Sfnp

There exists at least one deficient number in the interval $[n,n+(\log n{)}^2]$ for all sufficiently large n.Template:Sfnp

Related concepts

Template:Euler diagram numbers with many divisors.svg Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n.

Nicomachus was the first to subdivide numbers into deficient, perfect, or abundant, in his Introduction to Arithmetic (circa 100 CE). However, he applied this classification only to the even numbers.Template:Sfnp

See also

• Almost perfect number
• Amicable number
• Sociable number
• Superabundant number

References

• Template:Cite book
• Template:Cite journal
• Template:Cite book

External links

• The Prime Glossary: Deficient number
• Template:MathWorld
• Template:PlanetMath