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puzzles/project_euler/pe/p396.html
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puzzles/project_euler/pe/p396.html
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<!DOCTYPE html>
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<html lang="en">
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<meta name="author" content="Colin Hughes" />
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<title>Problem 396 - Project Euler</title>
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<h2>Weak Goodstein sequence</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 396</h3><span style="width:300px;color:#666;">Published on Sunday, 30th September 2012, 02:00 am; Solved by 302</span></div>
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<div class="problem_content" role="problem">
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<p>
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For any positive integer n, the <b>nth weak Goodstein sequence</b> {g<sub>1</sub>, g<sub>2</sub>, g<sub>3</sub>, ...} is defined as:
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<ul>
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<li> g<sub>1</sub> = <var>n</var>
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<li> for <var>k</var> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 1, g<sub><var>k</var></sub> is obtained by writing g<sub><var>k</var>-1</sub> in base <var>k</var>, interpreting it as a base <var>k</var> + 1 number, and subtracting 1.
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</ul>
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The sequence terminates when g<sub><var>k</var></sub> becomes 0.
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</p>
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<p>
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For example, the 6th weak Goodstein sequence is {6, 11, 17, 25, ...}:
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<ul>
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<li> g<sub>1</sub> = 6.
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<li> g<sub>2</sub> = 11 since 6 = 110<sub>2</sub>, 110<sub>3</sub> = 12, and 12 - 1 = 11.
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<li> g<sub>3</sub> = 17 since 11 = 102<sub>3</sub>, 102<sub>4</sub> = 18, and 18 - 1 = 17.
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<li> g<sub>4</sub> = 25 since 17 = 101<sub>4</sub>, 101<sub>5</sub> = 26, and 26 - 1 = 25.
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</ul>
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and so on.
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</p>
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<p>
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It can be shown that every weak Goodstein sequence terminates.
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</p>
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<p>
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Let G(<var>n</var>) be the number of nonzero elements in the <var>n</var>th weak Goodstein sequence.<br />
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It can be verified that G(2) = 3, G(4) = 21 and G(6) = 381.<br />
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It can also be verified that ΣG(<var>n</var>) = 2517 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_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 8.
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</p>
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<p>
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Find the last 9 digits of ΣG(<var>n</var>) 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_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 16.
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