Unitary perfect number


A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.

Examples

is a unitary perfect number, because 1, 3, 4, 5, 12, 15, and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first five, and only known, unitary perfect numbers are:
6, 60, 90, 87360, 146361946186458562560000
The respective sums of proper unitary divisors:
There are no odd unitary perfect numbers. This follows since one has 2d* dividing the sum of the unitary divisors of an odd number. One gets this because the sum of all the unitary divisors is a multiplicative function and one has the sum of the unitary divisors of a power of a prime pa is pa + 1 which is 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 odd prime factors.