If you run the contour walker for N=2, you will get a lot of hits for a sum of 2, for instance:
10,841,593,393 was the largest value found when I stopped the search.
Inspection reveals that these are all of the form (6n^3+1)^3-(6n^3-1)^3-(6n^2)^3. The above list starts with n = 2.
Clearly, it you take all the terms of a sum from the previous section (that yields 2) and multiply them by 'm', the sum will be 2m^3, so there is a simple series that can generate an infinite number of examples of any sum of this form.
(c) John Whitehouse 2021