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An integer $n\geq 1$ is said quasiperfect number if the sum of its positive divisors $\sigma(n)$ is equal to $2n+1$. See the Wikipedia Quasiperfect number.

The idea of this post is ask about the following equations, that I've created, I don't know if it is in the literature (I wrote other equations, see the Appendix). I was inspired in [1]. A question that should be more nice than mine is what about the radical of quasiperfect numbers (Luca and Pomerance studied the radical of perfect numbers in an article).

I've considered the following equations $$\operatorname{rad}(\sigma(n))=\operatorname{rad}(2n+1)\tag{1}$$ where $\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p$ is defined as the product of distinct prime numbers dividing the integer $m>1$, with the definition $\operatorname{rad}(1)=1$ (see the Wikipedia Radical of an integer), and also for integers $n>1$ $$\operatorname{rad}(\sigma(n)-1)=\operatorname{rad}(2n).\tag{2}$$

I got the following conjectures.

Conjecture 1. An integer $n\geq 1$ is a quasiperfect number if an only if satisfies $(1)$.

This is true for $1\leq n\leq 10^7$.

Conjecture 2. For integers $2<n$, there aren't solutions of the system $$\left. \begin{array}{l} \operatorname{rad}(\sigma(n))=\operatorname{rad}(2n+1)\\ \operatorname{rad}(\sigma(n)-1)=\operatorname{rad}(2n) \end{array} \right\}.$$

Question. My suspect is that Conjecture 1 is false, can you find a counterexample for Conjeture 1? I don't know if some deduction is possible for Conjecture 2, can you evoke what work can be done about it, that is about its veracity? Many thanks.

I don't know if the study of such equations improve the mathematical content of the genuine problem concerning quasiperfect numbers, or if these questions are interesting (feel free to add comments).

Appendix: If you are interested in these equations you can to explore it, also you can also to study systems of equations adding equations as $\sigma(\sigma(n))=\sigma(2n+1)$, $\varphi(\sigma(n))=\varphi(2n+1)$ or similar than $(2)$, where $\varphi(m)$ denotes the Euler's totient function.

References:

[1] Solutions of $rad(\sigma(m))=2rad(m)$, from this MathOverflow (February, 2014).

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  • $\begingroup$ I've removed the old Conjecture 2, editing since now I think that it is obvious. $\endgroup$
    – user142929
    Aug 5, 2019 at 13:05
  • $\begingroup$ About my comment that is in the literature an article of Luca and Pomerance studying the radical of perfect numbers but that I don't know nothing about the radical of quasiperfect numbers, I add that neither I do not know if it is possible to study the radical of other sequences concerning unsolved problems related to the sum of divisors function, for example what about the radical of amicable numbers (I say this if in case that it is interesting, feel free to explore/ask about it, if isn't in the literature, I'm not able to do it). $\endgroup$
    – user142929
    Aug 6, 2019 at 8:48

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