88 lines
5.4 KiB
HTML
88 lines
5.4 KiB
HTML
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<h2>Perfection Quotients</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 241</h3><span style="width:300px;color:#666;">Published on Saturday, 18th April 2009, 02:00 am; Solved by 545</span></div>
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<p>For a positive integer <var>n</var>, let σ(<var>n</var>) be the sum of all divisors of <var>n</var>, so e.g. σ(6) = 1 + 2 + 3 + 6 = 12.
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</p>
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<p>A perfect number, as you probably know, is a number with σ(<var>n</var>) = 2<var>n</var>.</p>
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<p><div style='text-align:left;'>
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<table>
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<tr>
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<td>Let us define the <b>perfection quotient</b> of a positive integer as</td><td><var>p</var>(<var>n</var>)</td><td>= </td>
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<td><div style='text-align:center;'>σ(<var>n</var>)<br /><img src='images/blackdot.gif' width='30' height='1' alt='' /><br /><var>n</var></div></td>
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<td>.</td>
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</tr>
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</table>
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</p>
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<!--
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<p>Let us define the <b>perfection quotient</b> of a positive integer as <var>p</var>(<var>n</var>) = <font "size=4"> <sup>σ(<var>n</var>)</sup>⁄<sub><var>n</var></sub></font>.</p>
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-->
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<p>Find the sum of all positive integers <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 10<sup>18</sup> for which <var>p</var>(<var>n</var>) has the form <var>k</var> + <sup>1</sup>⁄<sub>2</sub>, where <var>k</var> is an integer.</p>
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