103 lines
5.9 KiB
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103 lines
5.9 KiB
HTML
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<h2>A Modified Collatz sequence</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 277</h3><span style="width:300px;color:#666;">Published on Saturday, 6th February 2010, 01:00 am; Solved by 1670</span></div>
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A modified Collatz sequence of integers is obtained from a starting value a<sub>1</sub> in the following way:</p>
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<var>a<sub>n+1</sub></var> = <var>a<sub>n</sub></var>/3 if <var>a<sub>n</sub></var> is divisible by 3. We shall denote this as a large downward step, "D".</p>
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<p>
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<var>a<sub>n+1</sub></var> = (4<var>a<sub>n</sub></var> + 2)/3 if <var>a<sub>n</sub></var> divided by 3 gives a remainder of 1. We shall denote this as an upward step, "U".
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</p>
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<p>
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<var>a<sub>n+1</sub></var> = (2<var>a<sub>n</sub></var> - 1)/3 if <var>a<sub>n</sub></var> divided by 3 gives a remainder of 2. We shall denote this as a small downward step, "d".
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</p>
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<p>
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The sequence terminates when some <var>a<sub>n</sub></var> = 1.
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<p>
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Given any integer, we can list out the sequence of steps.<br />
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For instance if <var>a</var><sub>1</sub>=231, then the sequence {<var>a<sub>n</sub></var>}={231,77,51,17,11,7,10,14,9,3,1} corresponds to the steps "DdDddUUdDD".
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<p>
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Of course, there are other sequences that begin with that same sequence "DdDddUUdDD....".<br />
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For instance, if <var>a</var><sub>1</sub>=1004064, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.<br />
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In fact, 1004064 is the smallest possible <var>a</var><sub>1</sub> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 10<sup>6</sup> that begins with the sequence DdDddUUdDD.
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What is the smallest <var>a</var><sub>1</sub> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 10<sup>15</sup> that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?
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