# Problem 153: Investigating Gaussian Integers
As we all know the equation x2=-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 ≠
0 we call such integers "rational integers." A Gaussian integer is
called a divisor of a rational integer n if the result is also a
Gaussian integer. If for example we divide 5 by 1+2i we can simplify in
the following manner: Multiply numerator and denominator by the complex
conjugate of 1+2i: 1−2i. The result is . So 1+2i is a divisor of 5. Note
that 1+i is not a divisor of 5 because . 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 partSum s(n) of these divisors111 21, 1+i, 1-i, 25 31, 34 41, 1+i,
1-i, 2, 2+2i, 2-2i,413 51, 1+2i, 1-2i, 2+i, 2-i, 512 For divisors with
positive real parts, then, we have: . For 1 ≤ n ≤ 105, ∑
s(n)=17924657155. What is ∑ s(n) for 1 ≤ n ≤ 108?