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Steps in Euclid's algorithm

Problem 433

Published on Saturday, 22nd June 2013, 04:00 pm; Solved by 252

Let E(x0, y0) be the number of steps it takes to determine the greatest common divisor of x0 and y0 with Euclid's algorithm. More formally:
x1 = y0, y1 = x0 mod y0
xn = yn-1, yn = xn-1 mod yn-1
E(x0, y0) is the smallest n such that yn = 0.

We have E(1,1) = 1, E(10,6) = 3 and E(6,10) = 4.

Define S(N) as the sum of E(x,y) for 1 ≤ x,y ≤ N.
We have S(1) = 1, S(10) = 221 and S(100) = 39826.

Find S(5·106).