Is there a solution to this system of equations?

Question

Is there an integer solution to this system of equations?

$$\gcd(x, \sigma(x)) = 2x - \sigma(x) = \frac{x}{3} = \frac{\sigma(x)}{5}$$

Answer 1

Just looking at the back end of your equation, if $\frac{x}{3}=\frac{\sigma(x)}{5}$, then $\frac{\sigma(x)}{x}=\frac53$.

This would imply that $5x$ is an odd perfect number. (http://mcdanielabundancy.wdfiles.com/local--files/start/WeinerAbundDense.pdf)

This suggests that it would be hard to find such a number.

Answer 2

Just a partial answer to my question:

$x = {3003}^2$ "satisfies" the system of equations, save up to only

$$\gcd(x,\sigma(x)) = 2x - \sigma(x) = \frac{x}{3}.$$

This solution corresponds to the only known Descartes / spoof odd perfect number.

Edit: Well actually, $x = {3003}^2$ only satisfies up to $$\gcd(x,\sigma(x)) = 2x - \sigma(x) = 819 \neq \frac{x}{3}.$$

In general, by paw88789's answer, since $5x$ turns out to be an odd perfect number, then we have $$x > 2\cdot{{10}^{1499}}$$ by Ochem and Rao's result.

So, we have that $$\gcd(x,\sigma(x)) = \frac{x}{3} > \frac{2}{3}\cdot{{10}^{1499}}.$$

This makes a brute force search for such $x$ even harder.