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Problem 343

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Fractional Sequences

For any positive integer k, a finite sequence a_i of fractions x_i/y_i is defined by:
a₁ = 1/k and
a_i = (x_i-1+1)/(y_i-1-1) reduced to lowest terms for i>1.
When a_i reaches some integer n, the sequence stops. (That is, when y_i=1.)
Define f(k) = n.
For example, for k = 20:

1/20 → 2/19 → 3/18 = 1/6 → 2/5 → 3/4 → 4/3 → 5/2 → 6/1 = 6

So f(20) = 6.

Also f(1) = 1, f(2) = 2, f(3) = 1 and Σf(k³) = 118937 for 1 ≤ k ≤ 100.

Find Σf(k³) for 1 ≤ k ≤ 2×10⁶.

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分数序列

对于任意正整数k，由分数x_i/y_i组成的有限序列a_i按如下方式定义：
a₁ = 1/k，以及
对于i>1，a_i = (x_i-1+1)/(y_i-1-1)，并约分到最简形式。
当a_i为某个整数n时，序列结束。（换句话说，当y_i=1时序列结束。）
定义f(k) = n。
例如，对于k = 20：

1/20 → 2/19 → 3/18 = 1/6 → 2/5 → 3/4 → 4/3 → 5/2 → 6/1 = 6

所以f(20) = 6。

此外，已知f(1) = 1，f(2) = 2，f(3) = 1以及对于1 ≤ k ≤ 100，Σf(k³) = 118937。

对于1 ≤ k ≤ 2×10⁶，求Σf(k³)。

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