The inverse summation of coprime couples
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 q M
- p + q M
- p and q are coprime.
We also define S(N) as the sum of R(i) for 2 i N.
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.