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In this post we denote the sum of positive divisors function of an integer $n\geq 1$ as $$\sigma(n)=\sum_{1\leq d\mid n}d.$$

Then a prime of the form $2^p-1$ is called a Mersenne prime. These are related to the unsolved problem related to even perfect numbers. In this post I present two conjectures, with the hope to know if it is possible to deduce that these are rights as characterizations of Mersenne primes.

Conjecture 1. If $m\geq 1$ is an integer that satisfies

$$\sigma\left(\sigma\left(\sigma\left(\frac{m(m+1)}{2}\right)\right)\right)=(2m+1)\sigma\left(2m+1\right),\tag{1}$$ then $m$ is a Mersenne prime.

Conjecture 2. Let be $k\geq 1$ a fixed integer. If $m\geq 1$ is an integer satisfying

$$\sigma\left(\left(\frac{m(m+1)}{2}\right)^k\right)=\left(2\left(\frac{m+1}{2}\right)^k-1\right)\frac{m^{k+1}-1}{m-1}\tag{2}$$ then $m$ is a Mersenne prime.

Question. Are rights the previous conjectures? What work can be done about the veracity of these? Many thanks.

Also feel free, if you want, to add comments about these equations.

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  • $\begingroup$ I believe that also is true this very similar conjecture: Let be $k\geq 1$ a fixed integer, one has that an integer $s>3$ is a safe prime (sequence A005385 from the OEIS) if and only if $s$ satisfies $$\sigma\left(\left(\frac{s(s-1)}{2}\right)^k\right)=\frac{2}{s-3}\left(\left(\frac{s-1}{2}\right)^{k+1}-1\right)\frac{s^{k+1}-1}{s-1}.$$ $\endgroup$
    – user142929
    Commented Oct 2, 2019 at 22:54
  • $\begingroup$ And I think that also it is possible to write a similar formula than $(2)$ for Fermat primes. I would like to know the opinion of users to know if conjectures as Conjecture 2 can be potentially interesting in the study of Mersenne primes, and if to create versions of this conjecture for other constellations of primes as safe primes or Fermat primes can be interesting or useful in the future. $\endgroup$
    – user142929
    Commented Oct 2, 2019 at 23:31

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