Hi, I apologize if there is already an (obvious) answer to my question, but please bear with me for the moment as I find it hard to see a good way to answer this question:


In the same way that the Mersenne primes uniquely determine the even perfect numbers (i.e. in the "bijective" sense), do the Euler primes likewise uniquely determine the odd perfect numbers?


That is, while an Euler prime obviously maps to a single odd perfect number, it is not obvious to me that an odd perfect number could only map to a single Euler prime.


This is because:


(1) In the most general case, it is not known whether squares are friendly or solitary.


(2) It is not even known if the smallest possible Euler prime 5 does divide an odd perfect number.


(3) It is conceivable that two distinct numbers $P$ and $Q$ which satisfy a particular number-theoretic equation do not necessarily have to satisfy $\gcd(P, Q) = 1$.


I hope you guys can help me out on this one. Establishing this very last gap will provide a proof for the implication:


Assuming Sorli's Conjecture for OPNs is true, then there are no OPNs.



Perfect Numbers - On Mersenne and Euler Primes
I tried to get in touch with Dean Hickerson and here's what he got to say regarding the problem of determining the status of squares with respect to solitude or friendliness:



> 
> Notice that all of the squares from $1$ to $121$ are
>  solitary (since they satisfy $gcd(n, \sigma(n)) = 1$). (See [OEIS - A014567](http://oeis.org/A014567).)
>  Does my observation hold true in general?
> 
> 
> 


No. The smallest square for which $\gcd(n, \sigma(n))$ is not equal to $1$ is $196$: $\gcd(196, \sigma(196)) = 7$. (But it's easy to show that $196$ is solitary.)


In $1995$ I found a square that isn't solitary; I don't know if there are any smaller ones: $26334^2 = 693479556 = 2^2 3^4 7^2 11^2 19^2$. There are at least $5$ other numbers with the same abundancy index:


$$ 8640 = {2^6}\cdot{3^3}\cdot{5}$$


$$ 52416 = {2^6}\cdot{3^2}\cdot{7}\cdot{13}$$


$$ 71814642425856 = {2^{13}}\cdot{3^4}\cdot{11^3}\cdot{31}\cdot{43}\cdot{61}$$


$$ 2168446760665473024 = {2^{13}}\cdot{3^{10}}\cdot{11}\cdot{23}\cdot{43}\cdot{107}\cdot{3851}$$


$$ 5321505362711814144 = {2^{13}}\cdot{3^6}\cdot{11}\cdot{23}\cdot{43}\cdot{137}\cdot{547}\cdot{1093}$$



> 
> I had expected my conjecture to fail for even 
>  squares. Notwithstanding, have you also found
>  any odd squares which are not solitary?
> 
> 
> 


No, I haven't. It's easy to find odd squares for which $\gcd(n, \sigma(n))$ is not equal to $1$; e.g. if $n = {21}^2 = 441$ then $\gcd(n, \sigma(n)) = 3$. But $441$ is solitary.


Hence, it appears that the status of odd squares is still an open problem.


