106 lines
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106 lines
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<h2>Coresilience</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 245</h3><span style="width:300px;color:#666;">Published on Friday, 15th May 2009, 02:00 pm; Solved by 508</span></div>
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<div class="problem_content" role="problem">
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<p>We shall call a fraction that cannot be cancelled down a resilient fraction.<br /> Furthermore we shall define the resilience of a denominator, <var>R</var>(<var>d</var>), to be the ratio of its proper fractions that are resilient; for example, <var>R</var>(12) = <sup>4</sup>⁄<sub>11</sub>.</p>
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<div style='text-align:left;'>
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<table>
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<tr>
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<td>The resilience of a number <var>d</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 1 is then</td>
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<td><div style='text-align:center;'>φ(<var>d</var>)<br /><img src='images/blackdot.gif' width='36' height='1' alt='' /><br />
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<var>d</var> - 1</div></td><td>, where φ is Euler's totient function.</td>
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</tr>
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</table>
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<table>
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<tr>
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<td>We further define the <b>coresilience</b> of a number <var>n</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 1 as <var>C</var>(<var>n</var>)</td><td>= </td>
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<td><div style='text-align:center;'><var>n</var> - φ(<var>n</var>)<br />
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<img src='images/blackdot.gif' width='54' height='1' alt='' /><br />
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<var>n</var> - 1</div></td><td>.</td>
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</tr>
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</table><table>
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<tr><td>The coresilience of a prime <var>p</var> is <var>C</var>(<var>p</var>)</td>
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<td>= </td>
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<td><div style='text-align:center;'>1<br /><img src='images/blackdot.gif' width='34' height='1' alt='' /><br /><var>p</var> - 1</div></td><td>.</td>
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</tr>
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</table>
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<!--
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<p>The resilience of a number <var>d</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 1 is then <font "size=4"><sup>φ(<var>d</var>)</sup>⁄<sub>(<var>d</var>-1)</sub></font>, where φ is Euler's totient function.</p>
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<p>We further define the <b>coresilience</b> of a number <var>n</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 1 as <var>C</var>(<var>n</var>) = <font "size=4"><sup>(<var>n</var> - φ(<var>n</var>))</sup>⁄<sub>(<var>n</var> - 1)</sub></font>.
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</p>
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<p>The coresilience of a prime <var>p</var> is <var>C</var>(<var>p</var>) = <font "size=4"><sup>1</sup>⁄<sub>(<var>p</var> - 1)</sub></font>.</p>
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-->
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<p>Find the sum of all <b>composite</b> integers 1 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>n</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>11</sup>, for which <var>C</var>(<var>n</var>) is a <dfn title="A fraction with numerator 1">unit fraction</dfn>.
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