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. (A divisor $d$ of a number $n$ is a unitary divisor if $d$ and $\frac{n}{d}$ share no common factors.) Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.

The first few unitary perfect numbers are 6, 60, 90, 87360, 146361946186458562560000 (sequence A002827 in the OEIS).

There are no odd unitary perfect numbers. This follows since one has $2d*(n)$ dividing the sum of the unitary divisors of an odd number (where $d*(n)$ is the number of distinct prime divisors of $n$). 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 $p^a$ is $p^a+1$ 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’s not known whether or not there are infinitely many unitary perfect numbers.

Title	unitary perfect number
Canonical name	UnitaryPerfectNumber
Date of creation	2013-03-22 16:19:29
Last modified on	2013-03-22 16:19:29
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	4
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11A05