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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 divisors σ(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.

## 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, ... (sequence A005100 in 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 2 × 21, 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.
• An infinite number of both even and odd deficient numbers exist.
• All odd numbers with one or two distinct prime factors are deficient.
• All proper divisors of deficient or perfect numbers are deficient.
• There exists at least one deficient number in the interval ${\displaystyle [n,n+(\log n)^{2}]}$ 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).

## References

1. ^ Sándor et al (2006) p.108
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