Question about the Divisor Function/Sums and Project Euler

[Delta31415]

So I am kind of lost... I don't really know how to ask this.
Project Euler is a website that hosts multiple programming contests and I am interested in this problem
https://projecteuler.net/problem=608
but my question isn't truly about this problem but a more solution.

I know that the Divisor Function is
{\displaystyle \sigma _{x}(n)=\sum _{d\mid n}d^{x}\,\!}
my question is can I set the Divisor function equal to an algebraic function of value x and solve for the solutions
for example:

{\displaystyle \sigma _{x}(n)=\sum _{d\mid n}d^{x}\,\!} = (x^2+7)\ or\ (x+9)\ or\ even\ ln(x)
p.s I know that I can do this using programming and I have but I would like to be able to do it by hand as well.
I have recently been reading some elementary number theory textbooks but all they do is talk about primes and things such as the division algorithm, the divisor function isn't even mentioned.

Thanks for the help and I am sorry for being so confusing.
Edit: finally fixed the Latex

 

[willem2]

Number theory seems to be concerned with keeping x a fixed integer and varying n. Varying x while keeping n fixed is more a real analysis question. Even for
something as simple as \sigma_x(2) = ax + b you won't be able to get an answer using elementary functions for most values of a and b.. You could look up the lambert W function
https://en.wikipedia.org/wiki/Lambert_W_function#Solutions_of_equations (see example 1)

 

[Delta31415]

Number theory seems to be concerned with keeping x a fixed integer and varying n. Varying x while keeping n fixed is more a real analysis question. Even for
something as simple as \sigma_x(2) = ax + b you won't be able to get an answer using elementary functions for most values of a and b.. You could look up the lambert W function
https://en.wikipedia.org/wiki/Lambert_W_function#Solutions_of_equations (see example 1)

so I have been reading the wrong textbooks and should look into analytic number theory and real analysis and thxs for the link

Edit: I have looked at the link and most of it involves exponentials, so my question should I try to convert the divisor function into an integral or is this even possible

 

[Delta31415]

I just noticed a big mistake that I made and I cannot edit it now >_>

The X in the divisor function should be 1 as I am finding the sum of proper divisors and the algebraic variable should be n instead of x as x has a value of 1 in this case
{\displaystyle \sigma _{1}(n)=\sum _{d\mid n}d^{1}\,\!} = an+b

which would make it a more of a number theory question

 

[willem2]

{\displaystyle \sigma _{1}(n)=\sum _{d\mid n}d^{1}\,\!} = an+b

which would make it a more of a number theory question

Yes. If a =2 and b=0 these are the perfect numbers. If a>2 and b is 0, mutltiperfect numbers.
You probably need a different procedure for all different values of a and b. For example σ(n) = n+1 is valid if n is prime, σ(n) = n+2 is impossible, etc.
It's unknown wether there are odd perfect numbers, and for even perfect numbers you need to find mersenne primes. Techniques to find these or prove them impossible would probably apply for other values of a and b.