I am trying to find all of the answers to $r_2(n^2) = 420$, where $N < 10^{11}$. It is for finding lattice points on a circle with points $(0,0), (N,0), (0,N)$, and $(N,N)$. I am (pretty) sure that all of the following answers work:
359125, 469625, 612625, 718250, 781625, 866125, 933725, 939250, 1047625, 1119625, 1225250, 1288625, 1336625, 1366625
where $r_k(n)$ is the number of different squares $k$ (in this case, the sum of 2 different squares) that add up to $n$. (Source)
I have noticed that all of these are divisible by $5^2$, but I know that the answer to this problem ends in 309.
What kinds of numbers would get me 420 for this?