Sum of Squares
Consider equations of the form: $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.
For $N=65$ there are two solutions:
$a=1$, $b=8$ and $a=4$, $b=7$.
We call $S(N)$ the sum of the values of $a$ of all solutions of $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer.
Thus $S(65) = 1 + 4 = 5$.
Find $\sum S(N)$, for all squarefree $N$ only divisible by primes of the form $4k+1$ with $4k+1 \lt 150$.