82 lines
6.3 KiB
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82 lines
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<h2>Prime factors of <var>n</var><sup>15</sup>+1</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 421</h3><span style="width:300px;color:#666;">Published on Sunday, 31st March 2013, 04:00 am; Solved by 212</span></div>
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Numbers of the form <var>n</var><sup>15</sup>+1 are composite for every integer <var>n</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 1.<BR />
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For positive integers <var>n</var> and <var>m</var> let <var>s</var>(<var>n,m</var>) be defined as the sum of the <i>distinct</i> prime factors of <var>n</var><sup>15</sup>+1 not exceeding <var>m</var>.
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E.g. 2<sup>15</sup>+1 = 3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />11<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />331.<BR />
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So <var>s</var>(2,10) = 3 and <var>s</var>(2,1000) = 3+11+331 = 345.<BR /><BR />
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Also 10<sup>15</sup>+1 = 7<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />11<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />13<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />211<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />241<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />2161<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />9091.<BR />
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So <var>s</var>(10,100) = 31 and <var>s</var>(10,1000) = 483.<BR />
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Find ∑ <var>s</var>(<var>n</var>,10<sup>8</sup>) for 1 <img src='images/symbol_le.gif' width='10' height='12' 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;' /> 10<sup>11</sup>.
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