Lattice points on a circle

Let f(N) be the number of points with integer coordinates that are on a circle passing through (0,0), (N,0),(0,N), and (N,N).

It can be shown that f(10000) = 36.

What is the sum of all positive integers N ≤ 10¹¹ such that f(N) = 420 ?

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter