Editing Deficient number (section)

==Properties==
Since the aliquot sums of prime numbers equal 1, all [[prime number]]s are deficient.{{sfnp|Prielipp|1970|loc=Theorem 1, pp. 693–694}} More generally, all odd numbers with one or two distinct prime factors are deficient.  It follows that there are infinitely many [[odd number|odd]] deficient numbers.  There are also an infinite number of [[even number|even]] deficient numbers as all [[Power of two|powers of two]] have the sum ({{math|1 + 2 + 4 + 8 + ... + 2{{sup|''x''-1}} {{=}} 2{{sup|''x''}} - 1}}). The infinite family of numbers of form 2^(n - 1) * p^m where m > 0 and p is a prime > 2^n - 1 are also deficient.

More generally, all [[prime power]]s <math>p^k</math> are deficient, because their only proper divisors are <math>1, p, p^2, \dots, p^{k-1}</math> which sum to <math>\frac{p^k-1}{p-1}</math>, which is at most <math>p^k-1</math>.{{sfnp|Prielipp|1970|loc=Theorem 2, p. 694}}

All proper [[divisor]]s of deficient numbers are deficient.{{sfnp|Prielipp|1970|loc=Theorem 7, p. 695}}  Moreover, all proper divisors of [[perfect number]]s are deficient.{{sfnp|Prielipp|1970|loc=Theorem 3, p. 694}}

There exists at least one deficient number in the interval <math>[n, n + (\log n)^2]</math> for all sufficiently large ''n''.{{sfnp|Sándor|Mitrinović|Crstici|2006|p=108}}