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<h2>Alexandrian Integers</h2>
<h3>Problem 221</h3>
<div style="font-size:80%;color:#999;">Published on 13 December 2008 at 01:00 pm [Server Time]</div>
<div class="problem_content" role="problem">
<p>We shall call a positive integer <var>A</var> an "Alexandrian integer", if there exist integers <var>p</var>, <var>q</var>, <var>r</var> such that:</p>
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<var>A</var> = <var>p</var> · <var>q</var> · <var>r</var> and
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<tr><td>1</td></tr>
<tr><td class="overline"><var>A</var></td></tr>
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<td>=</td>
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<table class="frac">
<tr><td>1</td></tr>
<tr><td class="overline"><var>p</var></td></tr>
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</td>
<td>+</td>
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<table class="frac">
<tr><td>1</td></tr>
<tr><td class="overline"><var>q</var></td></tr>
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</td>
<td>+</td>
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<table class="frac">
<tr><td>1</td></tr>
<tr><td class="overline"><var>r</var></td></tr>
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<p>For example, 630 is an Alexandrian integer (<var>p</var> = 5, <var>q</var> = −7, <var>r</var> = −18).
In fact, 630 is the 6<sup>th</sup> Alexandrian integer, the first 6 Alexandrian integers being: 6, 42, 120, 156, 420 and 630.</p>
<p>Find the 150000<sup>th</sup> Alexandrian integer.</p>
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