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Problem 441


Problem 441


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.


互质数对的倒数和

对于整数M,定义R(M)是所有1/(p·q) 之和,其中整数对p和q满足如下条件:

  • 1 ≤ p < q ≤ M
  • p + q ≥ M
  • p和q互质。

对于2 ≤ i ≤ N,定义S(N)是所有R(i)的和。
可以验证S(2) = R(2) = 1/2,S(10) ≈ 6.9147以及S(100) ≈ 58.2962。

求S(107),并保留4位小数。