As we all know the equation $x^2=-1$ has no solutions for real $x$.
If we however introduce the imaginary number $i$ this equation has two solutions: $x=i$ and $x=-i$.
If we go a step further the equation $(x-3{)}^2=-4$ has two complex solutions: $x=3+2i$ and $x=3-2i$.
$x=3+2i$ and $x=3-2i$ are called each others' complex conjugate.
Numbers of the form $a+bi$ are called complex numbers.
In general $a+bi$ and $a-bi$ are each other's complex conjugate.

A Gaussian Integer is a complex number $a+bi$ such that both $a$ and $b$ are integers.
The regular integers are also Gaussian integers (with $b=0$).
To distinguish them from Gaussian integers with $b\ne 0$ we call such integers "rational integers."
A Gaussian integer $a+bi$ is called a divisor of a rational integer $n$ if the result ${\displaystyle \frac{n}{a+bi}}$ is also a Gaussian integer.
If for example we divide $5$ by $1+2i$ we can simplify ${\displaystyle \frac{5}{1+2i}}$ in the following manner:
Multiply numerator and denominator by the complex conjugate of $1+2i$: $1-2i$.
The result is ${\displaystyle \frac{5}{1+2i}}={\displaystyle \frac{5}{1+2i}}{\displaystyle \frac{1-2i}{1-2i}}={\displaystyle \frac{5(1-2i)}{1-(2i{)}^2}}={\displaystyle \frac{5(1-2i)}{1-(-4)}}={\displaystyle \frac{5(1-2i)}{5}}=1-2i$.
So $1+2i$ is a divisor of $5$.
Note that $1+i$ is not a divisor of $5$ because ${\displaystyle \frac{5}{1+i}}={\displaystyle \frac{5}{2}}-{\displaystyle \frac{5}{2}}i$.
Note also that if the Gaussian Integer $(a+bi)$ is a divisor of a rational integer $n$, then its complex conjugate $(a-bi)$ is also a divisor of $n$.

In fact, $5$ has six divisors such that the real part is positive: $\{1,1+2i,1-2i,2+i,2-i,5\}$.
The following is a table of all of the divisors for the first five positive rational integers:

$n$	Gaussian integer divisors with positive real part	Sum $s(n)$ of these divisors
$1$	$1$	$1$
$2$	$1,1+i,1-i,2$	$5$
$3$	$1,3$	$4$
$4$	$1,1+i,1-i,2,2+2i,2-2i,4$	$13$
$5$	$1,1+2i,1-2i,2+i,2-i,5$	$12$

For divisors with positive real parts, then, we have: $\sum _{n=1}^{5}s(n)=35$.

$\sum _{n=1}^{{10}^5}s(n)=17924657155$.

What is $\sum _{n=1}^{{10}^8}s(n)$?