What work can be done to study the solutions of $\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)$?

For integers $$n\geq 1$$ I denote the Euler's totient function as $$\varphi(n)$$ and the divisor function $$\sum_{1\leq d\mid n}d$$ as $$\sigma(n)$$, that are two well-known mulitplicative functions. We assume also the theory of odd perfect numbers, see if you want the corresponding section of the Wikipedia with title Perfect number.

It is easy to prove the following statement, on assumption that there exists an odd perfect number $$x$$.

Fact. If $$x$$ is an odd perfect number then $$\varphi\left(x^{\sigma(x)}\sigma(x)^x\right)=2^{x-1} x^{3x-1}\varphi(x)\tag{1}$$ holds.

Computational fact. For integers $$1\leq n\leq 5000$$, the only solution of $$(1)$$ is $$n=1$$. To see it, after some seconds, choose GP as language and evaluate next code (it is just a line written in Pari/GP) in the web page Sage Cell Server

for (x = 1, 5*10^3,if (eulerphi(x^(sigma(x))*(sigma(x))^x)==2^(x-1)*x^(3*x-1)*eulerphi(x), print(x)))

I believe that the following conjecture holds.

Conjecture. The only solution of our equation $$(1)$$ is the integer $$1$$.

Motivation for the post. My belief is that an interesting way (but my attempts were failed) to study the unsolved problem related to odd perfect numbers (that is if there exist any of them) should be to create intrincated/artificious equations similar than $$(1)$$ involving the sum of divisors functions and the Euler's totient function with the purpose to invoke inequalitites, asymptotics, heuristics or conjectures for these arithmetic functions (my belief is that the problem of odd perfect numbers is related to the distribution of prime numbers, thus maybe in the equations similar than $$(1)$$ that previously I've evoked should be required also that arise functions as the radical of an integer $$\operatorname{rad}(x)$$ or even the prime-counting function $$\pi(x)$$, both specialized for odd perfect numbers $$x$$).

Question. What work can be done to prove of refute previous conjecture, that the only solution of $$\varphi\left(n^{\sigma(n)}\sigma(n)^n\right)=2^{n-1} n^{3n-1}\varphi(n)$$ should be $$n=1$$? It is welcome unconditionally statements or heuristics, but also feel free to invoke conjectures if you can get some advanced statement. Many thanks.

Thus, as how it is perceived in the title of the post, previous Question is also an invitation to add remarkable statements about the nature of the solutions of $$(1)$$, if we are in the situation that the Question can not be solved.

Last remarks to to emphasize my ideas. What is saying myself thus previous Motivation and Question? That of couse I understand that the equation/characterization for odd perfect nubmers by means the equation $$\sigma(x)=2x$$ for odd integers $$x\geq 1$$ is easiest (to understand and study it) than others involving more arithmetic functions, but in my belief is that there exists a chance to get some statement for odd perfect numbers by the method to create more intrincated/artificious equations.

I think that my question is interesting, and I think that arises in a natural way when one tries to drop solutions like $$2^{2^{\lambda-1}-1}$$, that is the sequence A058891 from the On-Line Encyclopedia of Integer Sequences, for equations like this $$\varphi(x^x\sigma(x))=x^x\varphi(x).$$ See if you want the code

for (x = 1, 10^4,if (eulerphi((x^x)*sigma(x))==(x^x)*eulerphi(x), print(x)))

• Many thanks for the upvote. – user142929 Jul 19 at 13:40

In general, problems involving the composition of multiplicative functions are very hard to analyze. I don't see any specific way to approach this problem, and I'm skeptical that this is likely to be a fruitful direction. That said, I don't have any strong intuition of whether there will be non-odd perfect numbers satisfying this equation (aside from x=1), but my guess is that there will not be, because if x is not an OPN, then $$x^\sigma(x)$$ will have primes raised to very different powers then $$\sigma(x)^x$$ will, and numbers where $$x$$ and $$\sigma(x)$$ have the same set of distinct prime factors are rare. Turning this idea into a proof may be difficult.