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)))

  • $\begingroup$ Many thanks for the upvote. $\endgroup$ – user142929 Jul 19 at 13:40
  • $\begingroup$ As aside comment is that I believe that similar discussions are feasible in terms of the Dedekind psi function $\psi(n)$, in particular one should to have the following claim: Each odd perfect number $n$ satisfies $$\psi(n^{\sigma(n)}\sigma(n)^n )=3\cdot 2^{n-1}n^{3n-1}\psi(n).$$ And I cann't find any integer satisfying this equation over the segment $1\leq n\leq 1000$. A reformulation is using the equation $\psi(n^{\sigma(n)}\sigma(n)^n )=3\cdot 2^{n-1}n^{3n}\cdot\frac{\sigma(\text{rad}(n))}{\text{rad}(n)}$. For the definiton $\text{rad}(n)$ see the Wikipedia Radical of an integer. $\endgroup$ – user142929 Oct 30 at 12:56

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.

  • $\begingroup$ Many thanks for your answer. I've persuaded to myself (it is not a scientific procedure) that the problem of odd perfect numbers is related to Riemann hypothesis. My attempt thus is to create artificious equations with the intention to invoke some equivalence to the RH with the hope to get some useful information (in my imagination it is just transfer information from the odd perfect number problem to the RH problem). I think that due to the bound given by Ochem and Rao the problem of odd perfect numbers is inaccessible for the human mind, and that behind this problem there a physical meaning. $\endgroup$ – user142929 Jul 21 at 20:42
  • $\begingroup$ What connection are you seeing between odd perfect numbers and the Riemann Hypothesis, @user142929? Are you referring to the elementary criterion (i.e. an inequality) by Lagarias? $\endgroup$ – Jose Arnaldo Bebita-Dris Jul 30 at 22:41
  • $\begingroup$ I am persuaded myself that there is some relationship (and this, the persuasion, is outside the scientific field) @JoseArnaldoBebita-Dris There are in the literature more equivalences/issues related to the Riemann hypothesis similar than Lagarias theorem, because the formulations are using number theoretic functions or special sequences: highly composite numbers, Nicolas criterion, Robin's theorem. In my imagination, from my human size, my ignorance, and as a non-professional mathematicianIn, I ear that these are like a purring/murmur about the OPN. $\endgroup$ – user142929 Jul 31 at 8:44

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