Are all unitary perfect numbers divisible by 6? I recently learned that numbers with the property $\sigma(n)=2n$, where $\sigma(n)$ is the sum of the unitary divisors of $n$, are called unitary perfect numbers, where a divisor $d$ is a unitary divisor iff $d$ and $n/d$ are coprime.  According to Wikipedia, there are no odd unitary perfect numbers.  The $5$ known unitary perfect numbers as of today are $6$, $60$, $90$, $87360$, and $146361946186458562560000$, all of which are divisible by $6$. 
My question is must all unitary perfect numbers be divisible by $6$?
 A: If not, they must have  more than 7 prime factors.
Note that for such a number u,  2 is the product of a bunch of fractions of the form (1+p^f)/p^f, where p is a prime such that p^f exactly divides u.  It has been observed that one of those primes is 2, and if 3 is not a prime, then the largest of the fractions is 8/7, which means there are at least six distinct primes dividing the product, and further u has to be a multiple of 32. (One can refine this slightly when it is realized that the second largest fraction is 14/13, giving at least seven distinct primes, and one can go further with this.). One can't have a 6k-1 prime to an odd power, nor 2 to an odd power.  As one adds more constraints, one finds with elementary means that not having 3 as prime factor implies lots of prime factors, giving that 2 occurs to a high power.
Edit 2018.07.28 GRP
Here is a partial list of factors that would contribute the most and result in (a lower bound for) the smallest such unitary perfect number not divisible by 3: (8/7, 14/13, 20/19, 26/25, 32/31, 38/37, 44/43, 62/61, 68/67, 74/73, 80/79, 98/97, 104/103, 110/109, 122/121, 128/127, 140/139,   ...). Note that only squares of (6k-1) primes occur.  We can multiply with cancellation some of the larger terms to get  512/365 times 512/473 times 68/67 times 80/79 times 98/97 times smaller factors. Even with these 17 factors, one falls short of 5/3,  and at least 40 more factors are needed to approach 2.
End Edit 2018.07.28 GRP
Gerhard "Powering Through The Primary Constraints" Paseman, 2018.07.23.
