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Scary Sphere

Problem 360

Published on Sunday, 27th November 2011, 01:00 am; Solved by 413

Given two points (x1,y1,z1) and (x2,y2,z2) in three dimensional space, the Manhattan distance between those points is defined as

Let C(r) be a sphere with radius r and center in the origin O(0,0,0).
Let I(r) be the set of all points with integer coordinates on the surface of C(r).
Let S(r) be the sum of the Manhattan distances of all elements of I(r) to the origin O.

E.g. S(45)=34518.

Find S(1010).