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The inverse summation of coprime couples

Problem 441 Published on Sunday, 20th October 2013, 10:00 am; Solved by 276;
Difficulty rating: 65%

For an integer M, we define R(M) as the sum of 1/(p·q) for all the integer pairs p and q which satisfy all of these conditions:

  • 1 ≤ p < qM
  • p + qM
  • p and q are coprime.

We also define S(N) as the sum of R(i) for 2 ≤ iN.
We can verify that S(2) = R(2) = 1/2, S(10) ≈ 6.9147 and S(100) ≈ 58.2962.

Find S(107). Give your answer rounded to four decimal places.