91 lines
6.4 KiB
HTML
91 lines
6.4 KiB
HTML
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<h2>Fractional Sequences</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 343</h3><span style="width:300px;color:#666;">Published on Saturday, 18th June 2011, 04:00 pm; Solved by 736</span></div>
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<p>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:<br />
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a<sub>1</sub> = 1/<var>k</var> and<br />
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a<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.<br />
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When 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.)<br />
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Define f(<var>k</var>) = <var>n</var>. <br />
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For example, for <var>k</var> = 20:
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</p>
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<p>
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1/20 <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> 2/19 <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> 3/18 = 1/6 <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> 2/5 <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> 3/4 <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> 4/3 <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> 5/2 <img src='images/symbol_maps.gif' width='15' height='7' alt='→' border='0' style='vertical-align:middle;' /> 6/1 = 6
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<p>
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So f(20) = 6.
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<p>
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Also f(1) = 1, f(2) = 2, f(3) = 1 and Σf(<var>k</var><sup>3</sup>) = 118937 for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>k</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 100.
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<p>
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Find Σf(<var>k</var><sup>3</sup>) for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>k</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />10<sup>6</sup>.
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