https://ddlgitgzzhm.github.io/tags/phi/ Recent content in Phi on GzzhM Hugo -- gohugo.io en-us GzzhM Thu, 16 Jul 2026 00:00:00 +0000 https://ddlgitgzzhm.github.io/p/acwing-basic-0x11/ Thu, 16 Jul 2026 00:00:00 +0000 https://ddlgitgzzhm.github.io/p/acwing-basic-0x11/ <h2 id="欧拉函数">欧拉函数 </h2><p>什么是欧拉函数</p> <p>对于一个数 N, <code>phi(N) = x</code> 其中 <code>x</code> 代表 <code>1~n</code> 中与<code>n</code> 互质的个数</p> <p>互质的意思就是 <code>n 与 a</code> 没有公因子</p> <h3 id="算法思路">算法思路 </h3><p>我们知道一个数 <code>n = p1^a1 * p2^a2 ... pn^an</code></p> <p>即对于 <code>p1</code> 有 <code>x1</code> 个数在 <code>1~n</code>中不是互质的因为他是因子，那么不互质的个数就是 <code>y1 = n - n/p1</code> 算个倍数</p> <p>同理可得其他质因子的算法 从而推出 <code>phi = n * (1 - 1/p1) * (1- 1/p2)... (1-1/pn)</code></p> <h3 id="代码">代码 </h3><div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt"> 1 </span><span class="lnt"> 2 </span><span class="lnt"> 3 </span><span class="lnt"> 4 </span><span class="lnt"> 5 </span><span class="lnt"> 6 </span><span class="lnt"> 7 </span><span class="lnt"> 8 </span><span class="lnt"> 9 </span><span class="lnt">10 </span><span class="lnt">11 </span><span class="lnt">12 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-c++" data-lang="c++"><span class="line"><span class="cl"><span class="kt">int</span> <span class="nf">phi</span><span class="p">(</span><span class="kt">int</span> <span class="n">x</span><span class="p">)</span> <span class="p">{</span> </span></span><span class="line"><span class="cl"> <span class="kt">int</span> <span class="n">res</span> <span class="o">=</span> <span class="n">x</span> <span class="p">;</span> </span></span><span class="line"><span class="cl"> <span class="k">for</span><span class="p">(</span><span class="kt">int</span> <span class="n">i</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span> <span class="n">i</span> <span class="o">&lt;=</span> <span class="n">x</span><span class="o">/</span> <span class="n">i</span> <span class="p">;</span> <span class="n">i</span> <span class="o">++</span> <span class="p">)</span> <span class="p">{</span> </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl"> <span class="k">if</span><span class="p">(</span><span class="n">x</span> <span class="o">%</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="p">)</span> <span class="p">{</span> </span></span><span class="line"><span class="cl"> <span class="n">res</span> <span class="o">=</span> <span class="n">res</span> <span class="o">-</span> <span class="n">res</span><span class="o">/</span><span class="n">i</span><span class="p">;</span> </span></span><span class="line"><span class="cl"> <span class="k">while</span><span class="p">(</span><span class="n">x</span> <span class="o">%</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span> <span class="p">)</span> <span class="n">x</span><span class="o">/=</span><span class="n">i</span> <span class="p">;</span> </span></span><span class="line"><span class="cl"> <span class="p">}</span> </span></span><span class="line"><span class="cl"> <span class="p">}</span> </span></span><span class="line"><span class="cl"><span class="k">if</span><span class="p">(</span><span class="n">x</span> <span class="o">&gt;</span> <span class="mi">1</span><span class="p">)</span> <span class="n">res</span> <span class="o">=</span> <span class="n">res</span> <span class="o">-</span> <span class="n">res</span> <span class="o">/</span> <span class="n">x</span> <span class="p">;</span> </span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="n">res</span><span class="p">;</span> </span></span><span class="line"><span class="cl"><span class="p">}</span> </span></span></code></pre></td></tr></table> </div> </div><h3 id="时间复杂度">时间复杂度 </h3><p><code>sqrtN</code></p>