yaworsw/euler-manager

---
:id: 343
:name: Fractional Sequences
:url: https://projecteuler.net/problem=343
:content: "For any positive integer <var>k</var>, a finite sequence a<sub><var>i</var></sub>
  of fractions x<sub><var>i</var></sub>/y<sub><var>i</var></sub> is defined by:  \na<sub>1</sub>
  = 1/<var>k</var> and  \na<sub><var>i</var></sub> = (x<sub><var>i</var>-1</sub>+1)/(y<sub><var>i</var>-1</sub>-1)
  reduced to lowest terms for <var>i</var>\\>1.  \nWhen a<sub><var>i</var></sub> reaches
  some integer <var>n</var>, the sequence stops. (That is, when y<sub><var>i</var></sub>=1.)
  \ \nDefine f(<var>k</var>) = <var>n</var>.   \nFor example, for <var>k</var> = 20:\n\n1/20
  → 2/19 → 3/18 = 1/6 → 2/5 → 3/4 → 4/3 → 5/2 → 6/1 = 6\n\nSo f(20) = 6.\n\nAlso f(1)
  = 1, f(2) = 2, f(3) = 1 and Σf(<var>k</var><sup>3</sup>) = 118937 for 1 ≤ <var>k</var>
  ≤ 100.\n\nFind Σf(<var>k</var><sup>3</sup>) for 1 ≤ <var>k</var> ≤ 2×10<sup>6</sup>.\n\n"