89 lines
6.6 KiB
HTML
89 lines
6.6 KiB
HTML
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<h2>Squarefree factors</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 362</h3><span style="width:300px;color:#666;">Published on Sunday, 11th December 2011, 07:00 am; Solved by 278</span></div>
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Consider the number 54.<BR />
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54 can be factored in 7 distinct ways into one or more factors larger than 1:<BR />
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54, 2<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />27, 3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />18, 6<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />9, 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;' />6, 2<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;' />9 and 2<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;' />3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />3.<BR />
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If we require that the factors are all squarefree only two ways remain: 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;' />6 and 2<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;' />3<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />3.
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Let's call Fsf(<var>n</var>) the number of ways <var>n</var> can be factored into one or more squarefree factors larger than 1, so
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Fsf(54)=2.
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Let S(<var>n</var>) be <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' />Fsf(<var>k</var>) for <var>k</var>=2 to <var>n</var>.
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S(100)=193.
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<P>
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Find S(10 000 000 000).
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