Editing Unitary perfect number (section)

==Properties==
There are no [[parity (mathematics)|odd]] unitary perfect numbers. This follows since 2<sup>''d''*(''n'')</sup> divides the sum of the unitary divisors of an odd number ''n'', where ''d''*(''n'') is the number of distinct [[prime number|prime]] factors of ''n''. One gets this because the sum of all the unitary divisors is a [[multiplicative function]] and one has that the sum of the unitary divisors of a [[prime power]] ''p''<sup>''a''</sup> is ''p''<sup>''a''</sup>&nbsp;+&thinsp;1 which is [[parity (mathematics)|even]] for all odd primes ''p''. Therefore, an odd unitary perfect number must have only one distinct prime factor, and it is not hard to show that a power of prime cannot be a unitary perfect number, since there are not enough divisors. 

It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known.  A sixth such number would have at least nine distinct odd prime factors.<ref name=Wall1988>{{cite journal | last=Wall | first=Charles R. | title=New unitary perfect numbers have at least nine odd components | journal=[[Fibonacci Quarterly]] | volume=26 | number=4 | pages=312–317 | year=1988 | doi=10.1080/00150517.1988.12429611 | issn=0015-0517 | mr=967649 | zbl=0657.10003 }}</ref>