Deficient number

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 (1 + 2 + 4 + 8 + ... + 2^{x − 1} = 2^x − 1). The infinite family of numbers of form 2^{n − 1} × p^m where m > 0 and p is a prime > 2ⁿ − 1 are also deficient.

More generally, all prime powers ${\displaystyle p^k}$ are deficient, because their only proper divisors are ${\displaystyle 1,p,p^2,\dots ,p^{k-1}}$ which sum to ${\displaystyle \frac{p^k-1}{p-1}}$, which is at most ${\displaystyle p^k-1}$.

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 ${\displaystyle [n,n+(\log n{)}^2]}$ for all sufficiently large n.

Related concepts

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.