厄拉多塞筛法

厄拉多塞筛法

厄拉多塞筛法：

西元前250年，希腊数学家厄拉多塞(Eeatosthese)想到了一个非常美妙的质数筛法，减少了逐一检查每个数的的步骤，可以比较简单的从一大堆数字之中，筛选出质数来，这方法被称作厄拉多塞筛法(Sieve of Eeatosthese)。

具体操作：先将 2~n 的各个数放入表中，然后在2的上面画一个圆圈，然后划去2的其他倍数；第一个既未画圈又没有被划去的数是3，将它画圈，再划去3的其他倍数；现在既未画圈又没有被划去的第一个数 是5，将它画圈，并划去5的其他倍数……依次类推，一直到所有小于或等于 n 的各数都画了圈或划去为止。这时，表中画了圈的以及未划去的那些数正好就是小于 n 的素数。

其实，当你要画圈的素数的平方大于 n 时，那么后面没有划去的数都是素数，就不用继续判了。如下图：

C++代码如下：

<code class="language-c++ hljs d has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;"><span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> countPrimes(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> n) {    <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">bool</span> *Del = <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">new</span> <span class="hljs-built_in" style="color: rgb(102, 0, 102); box-sizing: border-box;">bool</span>[n]; <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 标记是否被划去</span>    Del[<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>] = <span class="hljs-literal" style="color: rgb(0, 102, 102); box-sizing: border-box;">false</span>;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>; i<n; ++i)        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(i%<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span> == <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>)            Del[i] = <span class="hljs-literal" style="color: rgb(0, 102, 102); box-sizing: border-box;">true</span>;  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 2的倍数全部划去</span>        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">else</span>            Del[i] = <span class="hljs-literal" style="color: rgb(0, 102, 102); box-sizing: border-box;">false</span>;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">3</span>; i<n; i+=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>)        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(!Del[i])  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 之后第一个未被划去</span>        {            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(i*i>n)  <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">break</span>;  <span class="hljs-comment" style="color: rgb(136, 0, 0); box-sizing: border-box;">// 当前素数的平方大于n，跳出循环</span>            <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> j=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>; i*j<n; ++j)                Del[i*j] = <span class="hljs-literal" style="color: rgb(0, 102, 102); box-sizing: border-box;">true</span>;        }    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> count = <span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">0</span>;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">for</span>(<span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">int</span> i=<span class="hljs-number" style="color: rgb(0, 102, 102); box-sizing: border-box;">2</span>; i<n; ++i)        <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">if</span>(!Del[i])            ++count;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">delete</span> [] Del;    <span class="hljs-keyword" style="color: rgb(0, 0, 136); box-sizing: border-box;">return</span> count;}</code><code class="language-c++ hljs d has-numbering" style="display: block; padding: 0px; color: inherit; box-sizing: border-box; font-family: 'Source Code Pro', monospace;font-size:undefined; white-space: pre; border-radius: 0px; word-wrap: normal; background: transparent;">转自：http://blog.csdn.net/lisonglisonglisong/article/details/45309651</code><ul class="pre-numbering" style="box-sizing: border-box; position: absolute; width: 50px; top: 0px; left: 0px; margin: 0px; padding: 6px 0px 40px; border-right-width: 1px; border-right-style: solid; border-right-color: rgb(221, 221, 221); list-style: none; text-align: right; background-color: rgb(238, 238, 238);"><li style="box-sizing: border-box; padding: 0px 5px;">1</li><li style="box-sizing: border-box; padding: 0px 5px;">2</li><li style="box-sizing: border-box; padding: 0px 5px;">3</li><li style="box-sizing: border-box; padding: 0px 5px;">4</li><li style="box-sizing: border-box; padding: 0px 5px;">5</li><li style="box-sizing: border-box; padding: 0px 5px;">6</li><li style="box-sizing: border-box; padding: 0px 5px;">7</li><li style="box-sizing: border-box; padding: 0px 5px;">8</li><li style="box-sizing: border-box; padding: 0px 5px;">9</li><li style="box-sizing: border-box; padding: 0px 5px;">10</li><li style="box-sizing: border-box; padding: 0px 5px;">11</li><li style="box-sizing: border-box; padding: 0px 5px;">12</li><li style="box-sizing: border-box; padding: 0px 5px;">13</li><li style="box-sizing: border-box; padding: 0px 5px;">14</li><li style="box-sizing: border-box; padding: 0px 5px;">15</li><li style="box-sizing: border-box; padding: 0px 5px;">16</li><li style="box-sizing: border-box; padding: 0px 5px;">17</li><li style="box-sizing: border-box; padding: 0px 5px;">18</li><li style="box-sizing: border-box; padding: 0px 5px;">19</li><li style="box-sizing: border-box; padding: 0px 5px;">20</li><li style="box-sizing: border-box; padding: 0px 5px;">21</li><li style="box-sizing: border-box; padding: 0px 5px;">22</li><li style="box-sizing: border-box; padding: 0px 5px;">23</li><li style="box-sizing: border-box; padding: 0px 5px;">24</li><li style="box-sizing: border-box; padding: 0px 5px;">25</li><li style="box-sizing: border-box; padding: 0px 5px;">26</li><li></li></ul>