81 lines
5.2 KiB
HTML
81 lines
5.2 KiB
HTML
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<h2>Totient Stairstep Sequences</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 337</h3><span style="width:300px;color:#666;">Published on Saturday, 7th May 2011, 10:00 pm; Solved by 303</span></div>
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<p>Let {a<sub>1</sub>, a<sub>2</sub>,..., a<sub><var>n</var></sub>} be an integer sequence of length <var>n</var> such that:</p>
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<ul>
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<li>a<sub>1</sub> = 6</li>
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<li>for all 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>i</var> < <var>n</var> : φ(a<sub><var>i</var></sub>) < φ(a<sub><var>i</var>+1</sub>) < a<sub><var>i</var></sub> < a<sub><var>i</var>+1</sub> <sup>1</sup></li>
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</ul>
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<p>Let S(<var>N</var>) be the number of such sequences with a<sub><var>n</var></sub> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>N</var>.<br />
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For example, S(10) = 4: {6}, {6, 8}, {6, 8, 9} and {6, 10}.<br />
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We can verify that S(100) = 482073668 and S(10 000) mod 10<sup>8</sup> = 73808307.</p>
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<p>Find S(20 000 000) mod 10<sup>8</sup>.</p>
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<p><sup>1</sup> φ denotes <b>Euler's totient function</b>.</p>
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