I am trying to characterize the solutions of the following equation involving the sum of divisors function $\sigma(n)$ and the Euler's totient function, denoted here as $\varphi(n)$.

The equation is $$\sigma\left(\frac{x(x+1)}{2}\right)=\left(2+3\varphi\left(\frac{x+2}{3}\right)\right)\left(1+3\varphi\left(\frac{x+2}{3}\right)\right).\tag{1}$$

My exploration is summarized in next claims, that I show after the question.

Question. Can you provide me a characterization of the solutions of our equation $$\sigma\left(\frac{y(y+1)}{2}\right)=\left(2+3\varphi\left(\frac{y+2}{3}\right)\right)\left(1+3\varphi\left(\frac{y+2}{3}\right)\right)$$ for positive integers $y\geq 1$? With the purpose to get such characterization, if a full characterization is impossible or very difficult, feel free to add some remarkable proposition about the solutions of our equation $(1)$. Many thanks.

Claim 1. If $x\geq 7$ is a Mersenne prime and $\frac{x+2}{3}$ is a Wagstaff prime (see this Wikipedia) then $(1)$ holds.

Remark. I've created previous equation $(1)$ thus with the intention to combine both definitions: the definition of Mersenne and Wagstaff primes.

Claim 2. Let $x\geq 7$ is a Mersenne prime solving $(1)$, then $\frac{x+2}{3}$ is a Wagstaff prime.

Sketch of proof. One has that $$3\varphi\left(\frac{x+2}{3}\right)=\frac{-3+\sqrt{1+4x(x+1)}}{2}$$ implies $\varphi((2^p+1)/3)=\frac{1}{3}(2^p+1)-1$.$\square$

Claim 3. A) If $\frac{x+2}{3}$ is a prime number such that the equation $(1)$ holds, then $\frac{x(x+1)}{2}$ is a perfect number. B) If $x\geq 7$ is prime being a solution of $(1)$, and we assume also that $\frac{x+2}{3}$ is a prime number then $\sigma(\frac{x+1}{2})=x.$

  • $\begingroup$ If some user is interested in this kind of equations I wrote the following conjecture that harmonizes with Puzzle 517, due to Firoozbakht from The Prime Puzzles & Problems Connection, by Carlos Rivera. Conjecture 1. If an integer $y\geq 1$ satisfies $3\varphi(y)+\sigma(3y-1)=3(3y-2)$ then $y$ is a Wagstaff prime. And since I was thinking in the context of the New Mersenne conjecture I wrote also next that also harmonizes with Firoozbakht's equation. Conjecture 2. If an integer $q\geq 7$ satisfies $2\varphi(2(q+1))+\sigma\left(3(q+1)/8\right)+1=3q$, then $q$ is a Mersenne number. $\endgroup$ – user243301 Feb 11 '18 at 19:27
  • $\begingroup$ Additionally as a companion of previous comment, if some user is interested I tried to get different equations capturing the form of those exponents $p$ in the New Mersenne conjecture, this Wikipedia, but my attempts were failed. $\endgroup$ – user243301 Feb 11 '18 at 19:32

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