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<span class="post-meta-item-text">Edited on</span>
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<a href="/categories/%E7%AE%97%E6%B3%95/" itemprop="url" rel="index"><span itemprop="name">算法</span></a>
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,
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<a href="/categories/%E7%AE%97%E6%B3%95/%E5%88%B7%E9%A2%98/" itemprop="url" rel="index"><span itemprop="name">刷题</span></a>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="http://acm.hdu.edu.cn/showproblem.php?pid=2841">http://acm.hdu.edu.cn/showproblem.php?pid=2841</a></p>
<p><strong>题目大意：</strong><br />
在平面<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span>两点之间的矩形中一共有<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">m</span></span></span></span>棵树，求站在点<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span>的人一共能看到多少棵没有被挡住的树？</p>
<p><strong>分析：</strong><br />
如果一棵树的坐标为<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，且<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>=</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g = gcd(x,y)\not=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> ，则<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mfrac><mi>x</mi><mi>g</mi></mfrac><mo separator="true">,</mo><mfrac><mi>y</mi><mi>g</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x',y') = (\frac{x}{g},\frac{y}{g})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.2311079999999999em;vertical-align:-0.481108em;"></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.481108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7475em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.446108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.481108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>，说明<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>在点<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span>到点<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x',y')</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>的线段延长线上，所以<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>会被挡住</p>
<p>那么只要枚举所有满足坐标<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>c</mi><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">gcd(x,y)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">c</span><span class="mord mathnormal">d</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>的点即可，基本的容斥过程</p>
<p><strong>代码：</strong></p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br></pre></td><td class="code"><pre><span class="line">#include <cstdio></span><br><span class="line">#include <cstring></span><br><span class="line">#include <algorithm></span><br><span class="line"></span><br><span class="line">using namespace std;</span><br><span class="line">typedef long long ll;</span><br><span class="line"></span><br><span class="line">const ll mod = 1e9+7;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">const int maxn = 100000+10;</span><br><span class="line">vector<int> ft[120005];</span><br><span class="line">bool isprime[120000];</span><br><span class="line"></span><br><span class="line">void init()</span><br><span class="line">{</span><br><span class="line"> for (int i = 2 ; i <= maxn ; i ++)</span><br><span class="line"> {</span><br><span class="line"> if (!isprime[i])</span><br><span class="line"> {</span><br><span class="line"> for (int j = i ; j <= maxn ; j += i)</span><br><span class="line"> {</span><br><span class="line"> isprime[j] = true;</span><br><span class="line"> ft[j].push_back(i);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">ll solve(int x,int sta,ll n)</span><br><span class="line">{</span><br><span class="line"> int pos = 0;</span><br><span class="line"> ll temp = 1;</span><br><span class="line"> while (sta)</span><br><span class="line"> {</span><br><span class="line"> if (sta&1)</span><br><span class="line"> temp *= ft[x][pos];</span><br><span class="line"> pos ++;</span><br><span class="line"> sta >>= 1;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> return n/temp;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">int m,n;</span><br><span class="line"></span><br><span class="line">int main(){</span><br><span class="line"> init();</span><br><span class="line"> int T,t=1;</span><br><span class="line"> scanf("%d",&T);</span><br><span class="line"> while (T--)</span><br><span class="line"> {</span><br><span class="line"> scanf("%d%d",&m,&n);</span><br><span class="line"></span><br><span class="line"> if (m>n)</span><br><span class="line"> swap(m,n);</span><br><span class="line"> ll ans = 0;</span><br><span class="line"> for (int i = 1 ; i <= n ; i ++)</span><br><span class="line"> {</span><br><span class="line"> ans += m;</span><br><span class="line"> for (int j = 1 ; j < (1<<ft[i].size()); j ++)</span><br><span class="line"> {</span><br><span class="line"> ll temp = solve(i,j,m);int cp = j,cnt = 0;</span><br><span class="line"> //printf("cp = %n\t",cp);</span><br><span class="line"> while (cp)</span><br><span class="line"> {</span><br><span class="line"> cp -= cp&(-cp);</span><br><span class="line"> cnt++;</span><br><span class="line"> }</span><br><span class="line"> //printf("cnt = %n\n",cnt);</span><br><span class="line"> if (cnt&1) ans -= temp;</span><br><span class="line"> else ans += temp;</span><br><span class="line"> }</span><br><span class="line"> //printf("%d\n",ans);</span><br><span class="line"> }</span><br><span class="line"> printf("%lld\n",ans);</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line"> return 0;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="http://acm.hdu.edu.cn/showproblem.php?pid=5778">http://acm.hdu.edu.cn/showproblem.php?pid=5778</a><br />
<strong>题目大意：</strong><br />
给定一个数x，求正整数<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">y\geq2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span>，使得满足以下条件：<br />
1.<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y-x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>的绝对值最小<br />
2.y的质因数分解式中每个质因数均恰好出现2次。</p>
<p><strong>简单分析：</strong><br />
y的质因数分解式中每个质因数均恰好出现2次，显然y是个完全平方数，但不是所有的完全平方数满足每个质因数恰好出现2次，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y-x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>绝对值最小的数，显然当<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mi>y</mi></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.33693999999999996em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.70306em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-2.66306em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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注1：第二次从<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\sqrt{x}+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.23972em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8002800000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">x</span></span></span><span style="top:-2.76028em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.23972em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>也满足，且答案更小，<br />
注2：由于<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">y\geq2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span>，还需要加特判当<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>≤</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">x\leq4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span>时，y都只能等于4，4为最小的满足条件的y值，此时当输出<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">4-x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span></p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><span class="line">#include<stdio.h></span><br><span class="line">#include<algorithm></span><br><span class="line">#include<iostream></span><br><span class="line">#include<string.h></span><br><span class="line">#include<math.h></span><br><span class="line">using namespace std;</span><br><span class="line"></span><br><span class="line">long long int x,y,temp;</span><br><span class="line"></span><br><span class="line">bool cha(long long kai)</span><br><span class="line">{</span><br><span class="line"> long long int i;</span><br><span class="line"> for(i=2;i*i<kai;i++)</span><br><span class="line"> {</span><br><span class="line"> if (kai%i==0)</span><br><span class="line"> {</span><br><span class="line"> kai/=i;</span><br><span class="line"> if (kai%i==0)</span><br><span class="line"> return true;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> return false;</span><br><span class="line">}</span><br><span class="line">int main()</span><br><span class="line">{</span><br><span class="line"> int t;</span><br><span class="line"> scanf("%d",&t);</span><br><span class="line"> while(t--)</span><br><span class="line"> {</span><br><span class="line"> scanf("%lld",&x);</span><br><span class="line"> long long kai1=(long long)sqrt(x);</span><br><span class="line"> long long kai2=kai1;</span><br><span class="line"> while(cha(kai1))</span><br><span class="line"> {</span><br><span class="line"> kai1=kai1-1;</span><br><span class="line"> }</span><br><span class="line"> y=abs(kai1*kai1-x);</span><br><span class="line"> kai2++;</span><br><span class="line"> while(cha(kai2))</span><br><span class="line"> {</span><br><span class="line"> kai2=kai2+1;</span><br><span class="line"> }</span><br><span class="line"> y=min(y,abs(kai2*kai2-x));</span><br><span class="line"> if (x<=4)</span><br><span class="line"> y = 4-x;</span><br><span class="line"> printf("%lld\n",abs(y));</span><br><span class="line"> }</span><br><span class="line"> return 0;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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<p>题目链接：<br />
BC #80 B Segment<br />
题意：<br />
Rivendell非常神,喜欢研究奇怪的问题.<br />
今天他发现了一个有趣的问题.找到一条线段<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">x+y=q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span>,<br />
令它和坐标轴在第一象限围成了一个三角形,然后画线连接了坐标原点和线段上坐标为整数的格点.<br />
请你找一找有多少点在三角形的内部且不是线段上的点,并将这个数对P取模后告诉他.<br />
输入p和q。q是质数且<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>≤</mo><mn>1018</mn></mrow><annotation encoding="application/x-tex">q≤1018</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">8</span></span></span></span>,<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>P</mi><mo>≤</mo><mn>1018</mn></mrow><annotation encoding="application/x-tex">1≤P≤1018</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">1</span><span class="mord">8</span></span></span></span><br />
分析：<br />
答案的公式是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>n</mi><mi>s</mi><mo>=</mo><mo stretchy="false">(</mo><mi>q</mi><mtext>−</mtext><mn>2</mn><mo stretchy="false">)</mo><mtext>∗</mtext><mo stretchy="false">(</mo><mi>q</mi><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">ans=(q−2)∗(q−1)/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">n</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord">−</span><span class="mord">2</span><span class="mclose">)</span><span class="mord">∗</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mord">/</span><span class="mord">2</span></span></span></span>;<br />
但是因为p和q的范围太大！直接乘的话会爆long long！！！<br />
所以需要手动写快速乘法，类似于快速幂的写法！！！</p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><span class="line">#include <iostream></span><br><span class="line">#include <cstdio></span><br><span class="line">#include <cstring></span><br><span class="line">#include <algorithm></span><br><span class="line">#include <climits></span><br><span class="line">#include <cmath></span><br><span class="line">using namespace std;</span><br><span class="line"></span><br><span class="line">int T;</span><br><span class="line">long long q,p;</span><br><span class="line"></span><br><span class="line">long long mul(long long a,long long b)</span><br><span class="line">{</span><br><span class="line"> long long res=0,tmp=a;</span><br><span class="line"> while(b){</span><br><span class="line"> if(b&1) res=(res+tmp)%p;</span><br><span class="line"> tmp=(tmp+tmp)%p;</span><br><span class="line"> b>>=1;</span><br><span class="line"> }</span><br><span class="line"> return res;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">int main()</span><br><span class="line">{</span><br><span class="line"> freopen("80Bin.txt","r",stdin);</span><br><span class="line"> scanf("%d",&T);</span><br><span class="line"> while(T--){</span><br><span class="line"> cin>>q>>p;</span><br><span class="line"> if(q==2){</span><br><span class="line"> printf("0\n");</span><br><span class="line"> continue;</span><br><span class="line"> }</span><br><span class="line"> long long t=(q-1)/2%p;</span><br><span class="line"> long long ans=mul(q-2,t);</span><br><span class="line"> cout<<ans<<endl;</span><br><span class="line"> }</span><br><span class="line"> return 0;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="http://acm.hdu.edu.cn/showproblem.php?pid=1251">http://acm.hdu.edu.cn/showproblem.php?pid=1251</a></p>
<p><strong>题目大意：</strong><br />
Ignatius最近遇到一个难题,老师交给他很多单词(只有小写字母组成,不会有重复的单词出现),现在老师要他统计出以某个字符串为前缀的单词数量(单词本身也是自己的前缀).</p>
<p><strong>分析：</strong><br />
读入单词直接插入即可，由于这里要查询的是以该前缀为起始的单词数量，所以在加单词的时候，不仅要在单词末尾加值，同时在路径上所有结点都应该价值，最后查询该单词的权值即可</p>
<p><strong>代码：</strong></p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br></pre></td><td class="code"><pre><span class="line">#include<cstdio></span><br><span class="line">#include<algorithm></span><br><span class="line">#include<iostream></span><br><span class="line">#include<cstring></span><br><span class="line">#include<cmath></span><br><span class="line">using namespace std;</span><br><span class="line"></span><br><span class="line">typedef long long ll;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">struct Trie{</span><br><span class="line"> Trie *next[26];</span><br><span class="line"> int val;</span><br><span class="line"> Trie(){</span><br><span class="line"> val = 0;</span><br><span class="line"> for (int i = 0 ; i < 26 ; i ++)</span><br><span class="line"> next[i] = NULL;</span><br><span class="line"> }</span><br><span class="line">};</span><br><span class="line"></span><br><span class="line">void addword(char *str,Trie* node)</span><br><span class="line">{</span><br><span class="line"> if (node->next[str[0]-'a']==NULL)</span><br><span class="line"> {</span><br><span class="line"> node->next[str[0]-'a'] = new Trie;</span><br><span class="line"> node = node->next[str[0]-'a'];</span><br><span class="line"> node->val ++;</span><br><span class="line"> }</span><br><span class="line"> else</span><br><span class="line"> {</span><br><span class="line"> node = node->next[str[0]-'a'];</span><br><span class="line"> node->val ++;</span><br><span class="line"> }</span><br><span class="line"> //printf("%c\t",*str);</span><br><span class="line"> str++;</span><br><span class="line"> if (*str)</span><br><span class="line"> addword(str,node);</span><br><span class="line"> else</span><br><span class="line"> return;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">int query(char *str ,Trie *node)</span><br><span class="line">{</span><br><span class="line"> //printf("%c\t",*str);</span><br><span class="line"> if (node->next[*str-'a']==NULL)</span><br><span class="line"> return 0;</span><br><span class="line"> else</span><br><span class="line"> node= node->next[*str-'a'];</span><br><span class="line"> ++str;</span><br><span class="line"> if (*str)</span><br><span class="line"> return query(str,node);</span><br><span class="line"> else</span><br><span class="line"> return node->val;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">char str[100];</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">int main()</span><br><span class="line">{</span><br><span class="line"> Trie *head = new Trie;</span><br><span class="line"> char ch;</span><br><span class="line"> int cnt ;</span><br><span class="line"> while (ch=getchar())</span><br><span class="line"> {</span><br><span class="line"> cnt = 0;</span><br><span class="line"> if (ch=='\n')</span><br><span class="line"> break;</span><br><span class="line"></span><br><span class="line"> while (ch!='\n')</span><br><span class="line"> {</span><br><span class="line"> str[cnt++] = ch;</span><br><span class="line"> ch = getchar();</span><br><span class="line"> }</span><br><span class="line"> str[cnt]=0;</span><br><span class="line"> addword(str,head);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> while (~scanf("%s",str))</span><br><span class="line"> {</span><br><span class="line"> int ans = query(str,head);</span><br><span class="line"> printf("%d\n",ans);</span><br><span class="line"> }</span><br><span class="line"> return 0;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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<a href="/2021/01/20/LightOJ%201234%20Harmonic%20Number%20(%E8%B0%83%E5%92%8C%E7%BA%A7%E6%95%B0%E6%B0%B4%E9%A2%98%EF%BC%89/" class="post-title-link" itemprop="url">LightOJ 1234 Harmonic Number (调和级数水题）</a>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="http://acm.hust.edu.cn/vjudge/contest/70017#problem/I">http://acm.hust.edu.cn/vjudge/contest/70017#problem/I</a><br />
<strong>题目大意：</strong><br />
求调和级数前n项的和，T个样例，(<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo lspace="0em" rspace="0em">≤</mo><mi>T</mi><mo lspace="0em" rspace="0em">≤</mo><mn>10000</mn><mtext>，</mtext><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>8</mn></msup></mrow><annotation encoding="application/x-tex">1{\leq}T{\leq}10000 ，1 ≤ n ≤ 10^8</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="mord">1</span><span class="mord"><span class="mrel">≤</span></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord"><span class="mrel">≤</span></span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord cjk_fallback">，</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span>)<br />
<strong>分析：</strong><br />
直接求肯定TLE，但是如果使用公式的话前几项精度不够，所以前<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup><mtext>或</mtext><mn>1</mn><msup><mn>0</mn><mn>7</mn></msup><mtext>项</mtext></mrow><annotation encoding="application/x-tex">10^6或10^7项</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mord cjk_fallback">或</span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span><span class="mord cjk_fallback">项</span></span></span></span><br />
<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mtext>（</mtext><mi>N</mi><mtext>）</mtext></mrow><annotation encoding="application/x-tex">O（N）</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord cjk_fallback">（</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mord cjk_fallback">）</span></span></span></span>暴力跑出，然后使用高精度的调和级数求和公式，具体可以搜索维基百科或者百度欧拉常数<br />
<a target="_blank" rel="noopener" href="http://baike.baidu.com/link?url=BWFVuV7oshbt5k7Z2HhvmV84MlXGg2bBE0_MJsQ9ZOJLI8o773s5-Z6k6xK7csGekpFwn0kn539eYbgY-lUDeq">http://baike.baidu.com/link?url=BWFVuV7oshbt5k7Z2HhvmV84MlXGg2bBE0_MJsQ9ZOJLI8o773s5-Z6k6xK7csGekpFwn0kn539eYbgY-lUDeq</a><br />
最后的三个公式是表示欧拉常数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><msub><mi>H</mi><mi>n</mi></msub><mo>−</mo><mfrac><mrow><mi>l</mi><mi>n</mi><mtext> </mtext><mi>n</mi><mo>+</mo><mi>l</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>6</mn><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>30</mn><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">y = H_n - \frac{ln{\space}n+ln(n+1)}{2} - \frac{1}{6n(n+1)} + \frac{1}{30n^2(n+1)^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.355em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mord mathnormal mtight">n</span><span class="mord mtight"><span class="mspace mtight"><span class="mtight"> </span></span></span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mord mathnormal mtight">n</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.365108em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span><span class="mord mathnormal mtight">n</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.365108em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mtight">0</span><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463142857142857em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>H</mi><mi>n</mi></msub><mtext>为调和级数前</mtext><mi>n</mi><mtext>项的和</mtext><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H_n 为调和级数前n项的和)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">调</span><span class="mord cjk_fallback">和</span><span class="mord cjk_fallback">级</span><span class="mord cjk_fallback">数</span><span class="mord cjk_fallback">前</span><span class="mord mathnormal">n</span><span class="mord cjk_fallback">项</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">和</span><span class="mclose">)</span></span></span></span></p>
<p><strong>代码：</strong></p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br></pre></td><td class="code"><pre><span class="line">#include <iostream></span><br><span class="line">#include <cstdio></span><br><span class="line">#include <cstdlib></span><br><span class="line">#include <algorithm></span><br><span class="line">#include <cstring></span><br><span class="line">#include <string></span><br><span class="line">#include <vector></span><br><span class="line">#include <cmath></span><br><span class="line">#include <set></span><br><span class="line">#define MOD 998244353</span><br><span class="line">using namespace std;</span><br><span class="line">typedef unsigned long long ull;</span><br><span class="line">typedef long long ll;</span><br><span class="line"></span><br><span class="line">/*---------------------------head files----------------------------------*/</span><br><span class="line">const double euler = 0.577215664901532860606512090082;</span><br><span class="line">double arr[1000009];</span><br><span class="line"></span><br><span class="line">void init()</span><br><span class="line">{</span><br><span class="line"> arr[1] = 1.0;</span><br><span class="line"> for (int i = 2 ; i <= 1000000 ; i ++)</span><br><span class="line"> {</span><br><span class="line"> arr[i] = arr[i-1] + 1.0 / i ;</span><br><span class="line"> }</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">int main()</span><br><span class="line">{</span><br><span class="line"> init();</span><br><span class="line"> int T,k=1;</span><br><span class="line"> ll n;</span><br><span class="line"> scanf("%d",&T);</span><br><span class="line"> while (T--)</span><br><span class="line"> {</span><br><span class="line"> scanf("%lld",&n);</span><br><span class="line"> printf("Case %d: ",k++);</span><br><span class="line"> if (n<=1000000)</span><br><span class="line"> printf("%.8f\n",arr[n]);</span><br><span class="line"> else</span><br><span class="line"> {</span><br><span class="line"> double ans = (log(n) + log(n+1))/2.0;</span><br><span class="line"> double x1 = 1.0;</span><br><span class="line"> x1 = x1 / 6.0 / n / (n+1);</span><br><span class="line"> double x2 = x1 ;</span><br><span class="line"> x2 = x2 /5.0 / n / (n+1);</span><br><span class="line"> ans = ans - x1 + x2 ;</span><br><span class="line"> printf("%.10f\n",ans+euler);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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<a href="/2021/01/08/UVA%2011752%20%20The%20Super%20Powers%20%20%20%EF%BC%88%E9%9B%86%E5%90%88%E6%89%93%E8%A1%A8%EF%BC%89/" class="post-title-link" itemprop="url">UVA 11752 The Super Powers （集合打表）</a>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="http://acm.hust.edu.cn/vjudge/contest/70017#problem/Z">http://acm.hust.edu.cn/vjudge/contest/70017#problem/Z</a></p>
<p><strong>题目大意：</strong><br />
有些数既是一个数的p次方，又是另一个数的q次方，称这样的数为Super Power，没有输入，输出所有这样的数</p>
<p><strong>分析：</strong><br />
显而易见，满足可以写成两个以上的数不同次方的数，它的次方数必然是个合数，同时由于最小的合数是4，所以最大的底数是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow></mrow><mn>4</mn></msup><msqrt><mrow><msup><mn>2</mn><mn>64</mn></msup><mo>−</mo><mn>1</mn></mrow></msqrt><mo><</mo><msup><mn>2</mn><mn>16</mn></msup></mrow><annotation encoding="application/x-tex">^4\sqrt{2^{64}-1}<2^{16}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.12661100000000003em;"></span><span class="mord"><span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.913389em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span></span></span><span style="top:-2.873389em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.12661100000000003em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span><br />
至多65536个数，每个数至多64次幂，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(10^6)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，不会T<br />
为了防止重复，最好使用集合来存数，最后用迭代器输出集合，注意循环从2开始的话需要在一开始向集合里插入一个1</p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br></pre></td><td class="code"><pre><span class="line">#include<iostream></span><br><span class="line">#include<cstdio></span><br><span class="line">#include<cmath></span><br><span class="line">#include<cstring></span><br><span class="line">#include<map></span><br><span class="line">#include<set></span><br><span class="line">#include <algorithm></span><br><span class="line">using namespace std;</span><br><span class="line">typedef long long ll;</span><br><span class="line">typedef unsigned long long ull;</span><br><span class="line">const int maxn = 65;</span><br><span class="line">ll notprime[maxn];</span><br><span class="line">int tot ;</span><br><span class="line">bool num[maxn];</span><br><span class="line">set<ull> ans;</span><br><span class="line"></span><br><span class="line">ull quickpow(ll a,ll n)</span><br><span class="line">{</span><br><span class="line"> ull ans=1;</span><br><span class="line"> while(n)</span><br><span class="line"> {</span><br><span class="line"> if (n&1)</span><br><span class="line"> ans = ans * a;</span><br><span class="line"> a = a * a ;</span><br><span class="line"> n >>= 1;</span><br><span class="line"> }</span><br><span class="line"> return ans;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">void init()</span><br><span class="line">{</span><br><span class="line"> int i,j;</span><br><span class="line"> for ( int i = 2; i <= maxn ; i ++ )</span><br><span class="line"> if (num[i])</span><br><span class="line"> notprime[++tot]=i;</span><br><span class="line"> else</span><br><span class="line"> for (int j = 2 *i ; j <= maxn ; j += i )</span><br><span class="line"> num[j]=true;</span><br><span class="line"> ans.insert(1);</span><br><span class="line"> for (int i = 2 ; i < (1<<16);i ++)</span><br><span class="line"> {</span><br><span class="line"> int maxx = ceil(64*log(2.0)/log(i*1.0));</span><br><span class="line"> for (int j = 4 ; j < maxx ; j ++)</span><br><span class="line"> {</span><br><span class="line"> if (num[j])</span><br><span class="line"> ans.insert(quickpow(i,j));</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">int main()</span><br><span class="line">{</span><br><span class="line"> init();</span><br><span class="line"> int n;</span><br><span class="line"> for (auto i : ans)</span><br><span class="line"> {</span><br><span class="line"> printf("%llu\n",i);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line">}</span><br><span class="line"></span><br></pre></td></tr></table></figure>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="http://acm.hdu.edu.cn/showproblem.php?pid=6050">http://acm.hdu.edu.cn/showproblem.php?pid=6050</a></p>
<p><strong>题目大意：</strong><br />
对<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">F_{x,y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>给出三个定义式，求<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mi>m</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">%</mi><mn>1000000007</mn></mrow><annotation encoding="application/x-tex">F_{m,1}\%1000000007</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord">%</span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span><span class="mord">7</span></span></span></span>之后的值</p>
<p><strong>分析：</strong><br />
根据$F_{1,i} = F_{1,i-1} + 2*F_{1,i-2} $可以看出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>1</mn><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>−</mo><mn>2</mn><msub><mi>F</mi><mrow><mn>1</mn><mo separator="true">,</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_{1,i} - 2F_{1,i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>成等比，然后迭代即可求得<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>1</mn><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">F_{1,i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>的通项，然后求和便是<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_{2,1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>的值，之后发现，然后再对第二行的前n个数求和，得到<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_{3,1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>即可发现答案，或者打表寻找在不同的n下，从<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_{2,1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>到<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_{3,1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>及后面多个数的变化即可发现</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>F</mi><mrow><mi>m</mi><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>（</mtext><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∗</mo><msub><mi>F</mi><mrow><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mtext> </mtext><mi>n</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∗</mo><msub><mi>F</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>S</mi><mrow><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mtext> </mtext><mi>n</mi><mtext> </mtext><mi>i</mi><mi>s</mi><mtext> </mtext><mi>o</mi><mi>d</mi><mi>d</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">F_{m,1} =\begin{cases}
&（2^n-1)*F_{2,1} \space\space\space n \space is\space even\\
& (2^n-1)*F_{m-1,1}-S_{1,n-1} \space\space\space n\space is \space odd\end{cases}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord cjk_fallback">（</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord mathnormal">n</span><span class="mspace"> </span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mspace"> </span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord mathnormal">n</span><span class="mspace"> </span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mspace"> </span><span class="mord mathnormal">o</span><span class="mord mathnormal">d</span><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>n为奇数时减去的常数也可以由打表找出，打表发现<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n=3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span>时，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mn>2</mn><mn>3</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∗</mo><msub><mi>F</mi><mrow><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">F_{3,1} = (2^3 -1)*F_{2,1} - 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span> , <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n=5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span></span></span></span>时，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mrow><mn>3</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mn>2</mn><mn>5</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∗</mo><msub><mi>F</mi><mrow><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">F_{3,1} = (2^5-1)*F_{2,1} - 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span></span></span></span> ,而<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">n=7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">7</span></span></span></span>时，常数为42，可以发现<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msub><mo>=</mo><mn>4</mn><mo>∗</mo><msub><mi>C</mi><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo stretchy="false">]</mo><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msub><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">C_{(n-1)/2} = 4*C_{(n-3]/2}+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.03853em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span><span class="mord mtight">/</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.03853em;vertical-align:-0.3551999999999999em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.34480000000000005em;"><span style="top:-2.5198em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">3</span><span class="mclose mtight">]</span><span class="mord mtight">/</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3551999999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span>继续拆解构造数列通项式即可得到常数的表达式，然后快速幂即可得到答案，中间分数用逆元处理</p>
<p><strong>代码：</strong></p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><span class="line">#include <bits/stdc++.h></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">using namespace std;</span><br><span class="line"></span><br><span class="line">typedef long long ll;</span><br><span class="line"></span><br><span class="line">const int maxn = 2e5+200;</span><br><span class="line">const int mod = 1e9+7;</span><br><span class="line"></span><br><span class="line">ll n,m;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">ll f[200],arr[20][1000];</span><br><span class="line"></span><br><span class="line">ll quickpow(ll a,ll n)</span><br><span class="line">{</span><br><span class="line"> ll res = 1;</span><br><span class="line"> while (n)</span><br><span class="line"> {</span><br><span class="line"> if (n&1) res = res*a%mod;</span><br><span class="line"> a = a*a %mod;</span><br><span class="line"> n >>= 1;</span><br><span class="line"> }</span><br><span class="line"> return res;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"></span><br><span class="line">int main() {</span><br><span class="line"> int T,t=1;</span><br><span class="line"> ll rst = quickpow(3,mod-2);</span><br><span class="line"> ll rss = quickpow(2,mod-2);</span><br><span class="line"> scanf("%d",&T);</span><br><span class="line"></span><br><span class="line"> while (T--)</span><br><span class="line"> {</span><br><span class="line"> scanf("%lld%lld",&n,&m);</span><br><span class="line"> ll tnp = quickpow(2,n);</span><br><span class="line"> ll fsone = ((tnp*2-2+(n&1))*rst)%mod , p =tnp - 1;</span><br><span class="line"> ll rs = quickpow(p-1,mod-2);</span><br><span class="line"> if (m==1)</span><br><span class="line"> printf("1\n");</span><br><span class="line"> else if (n&1)</span><br><span class="line"> {</span><br><span class="line"> ll d = (tnp-2)%mod*rst%mod;</span><br><span class="line"> ll ans = (fsone-d*rs%mod+mod)%mod*quickpow(p,m-2)%mod+d*rs%mod;</span><br><span class="line"> printf("%lld\n",ans%mod);</span><br><span class="line"> }</span><br><span class="line"> else</span><br><span class="line"> {</span><br><span class="line"> ll ans = fsone*quickpow(tnp-1,m-2)%mod;</span><br><span class="line"> printf("%lld\n",ans);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> return 0;</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="https://vjudge.net/contest/70325#problem/A">https://vjudge.net/contest/70325#problem/A</a></p>
<p><strong>题目大意：</strong><br />
给定两个数字串，第一个为待匹配串，第二个为模式串，求模式串第一个在待匹配串中出现的下标，没有则返回-1</p>
<p><strong>分析：</strong><br />
简单修改KMP模板即可应用到数字串匹配上</p>
<p><strong>代码：</strong></p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br></pre></td><td class="code"><pre><span class="line">#include <cstdio></span><br><span class="line">#include <cstdlib></span><br><span class="line">#include <iostream></span><br><span class="line">#include <cstring></span><br><span class="line">#include <algorithm></span><br><span class="line">using namespace std;</span><br><span class="line"></span><br><span class="line">int a[1000009],b[10009];</span><br><span class="line">int n,m,Next[10009];</span><br><span class="line">int pos;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">void cover()</span><br><span class="line">{</span><br><span class="line"> int i , j = 0;</span><br><span class="line"> Next[0] = Next[1] = 0;</span><br><span class="line"> for (int i = 1; i < m ; i ++)</span><br><span class="line"> {</span><br><span class="line"> while (j&&b[i]!=b[j]) j = Next[j];</span><br><span class="line"> if (b[i]==b[j]) ++j;</span><br><span class="line"> Next[i+1] = j;</span><br><span class="line"> }</span><br><span class="line"> return ;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line">int kmp()</span><br><span class="line">{</span><br><span class="line"> int j = 0;</span><br><span class="line"> for (int i = 0 ; i < n ; i ++)</span><br><span class="line"> {</span><br><span class="line"> while (j&&a[i]!=b[j]) j = Next[j];</span><br><span class="line"> if (a[i]==b[j]){</span><br><span class="line"> if (j==m-1)</span><br><span class="line"> return i - j + 1;</span><br><span class="line"> ++j;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> return -1;</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">int main(){</span><br><span class="line"> int T;</span><br><span class="line"> scanf("%d",&T);</span><br><span class="line"> while (T--)</span><br><span class="line"> {</span><br><span class="line"> scanf("%d%d",&n,&m);</span><br><span class="line"> for (int i = 0 ; i < n ; i ++)</span><br><span class="line"> scanf("%d",&a[i]);</span><br><span class="line"> for (int i = 0 ; i < m ; i ++)</span><br><span class="line"> scanf("%d",&b[i]);</span><br><span class="line"> cover();</span><br><span class="line"> printf("%d\n",kmp());</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure>
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<p><strong>题目链接：</strong><br />
<a target="_blank" rel="noopener" href="http://poj.org/problem?id=3744">http://poj.org/problem?id=3744</a></p>