91 lines
5.2 KiB
HTML
91 lines
5.2 KiB
HTML
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<h2>Totient Chains</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 214</h3><span style="width:300px;color:#666;">Published on Saturday, 25th October 2008, 02:00 pm; Solved by 2861</span></div>
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<p>Let φ be Euler's totient function, i.e. for a natural number <var>n</var>,
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φ(<var>n</var>) is the number of <var>k</var>, 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;' /> <var>n</var>, for which gcd(<var>k</var>,<var>n</var>) = 1.</p>
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<p>By iterating φ, each positive integer generates a decreasing chain of numbers ending in 1.<br />
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E.g. if we start with 5 the sequence 5,4,2,1 is generated.<br />
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Here is a listing of all chains with length 4:</p>
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<div style="text-align:right; margin-right:350px;">
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5,4,2,1<br />
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7,6,2,1<br />
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8,4,2,1<br />
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9,6,2,1<br />
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10,4,2,1<br />
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12,4,2,1<br />
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14,6,2,1<br />
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18,6,2,1</div>
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<p>Only two of these chains start with a prime, their sum is 12.</p>
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<p>What is the sum of all primes less than 40000000 which generate a chain of length 25?</p>
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