Deficient number

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).

Since the aliquot sums of prime numbers equal 1, all prime numbers are deficient.

[1] More generally, all odd numbers with one or two distinct prime factors are deficient.

It follows that there are infinitely many odd deficient numbers.

[3] Moreover, all proper divisors of perfect numbers are deficient.

[4] There exists at least one deficient number in the interval

for all sufficiently large n.[5] 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).