{"id":1672,"date":"2021-03-18T10:57:18","date_gmt":"2021-03-18T02:57:18","guid":{"rendered":"http:\/\/47.101.202.111\/?p=1672"},"modified":"2021-10-04T22:40:28","modified_gmt":"2021-10-04T14:40:28","slug":"algorithm-prime-enumeration","status":"publish","type":"post","link":"http:\/\/139.196.114.170\/?p=1672","title":{"rendered":"\u7ebf\u6027\u65f6\u95f4\u679a\u4e3e\u7d20\u6570 – \u6570\u8bba\u65b9\u6cd5"},"content":{"rendered":"

\u7efc\u8ff0<\/h2>\n

\u679a\u4e3e\u533a\u95f4 $\\scriptsize [l, h]$ \u5185\u7684\u6240\u6709\u7d20\u6570\u95ee\u9898\uff0c\u65b9\u6cd5\u53ca\u65f6\u95f4\u590d\u6742\u5ea6\u5982\u4e0b<\/p>\n\n\n\n\n\n\n\n
\u65b9\u6cd5<\/th>\n\u601d\u60f3<\/th>\n\u65f6\u95f4<\/th>\n\u7a7a\u95f4<\/th>\n\u7d20\u6570\u8868<\/th>\n<\/tr>\n<\/thead>\n
\u679a\u4e3e\u8bd5\u9664<\/td>\n\u9010\u4e2a\u9664\u4f59\u5224\u65ad\u7d20\u6570<\/td>\n$\\scriptsize O(n\\sqrt n)$<\/td>\n$\\scriptsize O(n)$<\/td>\n\u4ec5\u6709\u7d20\u6570\u5224\u522b\u8868<\/td>\n<\/tr>\n
E-\u7b5b\u6cd5<\/td>\n\u5254\u9664\u6240\u6709\u8d28\u56e0\u5b50\u679a\u4e3e\u7684\u5408\u6570<\/td>\n$\\scriptsize O(n\\log \\log n)$<\/td>\n$\\scriptsize O(n)$<\/td>\n\u4ec5\u6709\u7d20\u6570\u5224\u522b\u8868<\/td>\n<\/tr>\n
\u6b27\u62c9\u7b5b\u6cd5<\/td>\n\u5254\u9664\u6700\u5c0f\u8d28\u56e0\u5b50\u679a\u4e3e\u7684\u5408\u6570<\/td>\n$\\scriptsize O(n)$<\/td>\n$\\scriptsize O(2n)$<\/td>\n\u5224\u522b\u8868\u548c\u7eaf\u7d20\u6570\u8868<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u679a\u4e3e\u8bd5\u9664<\/h3>\n

\u679a\u4e3e\u533a\u95f4\u4e2d\u6bcf\u4e2a\u6570\u5e76\u5224\u65ad\u7d20\u6570\uff1a\u679a\u4e3e\u4ece $\\scriptsize 2$ \u5230 $\\scriptsize n$ \u7684\u6240\u6709\u56e0\u5b50\uff0c\u6c42\u6a21\u5224\u65ad\u662f\u5426\u4e3a\u6240\u7ed9\u6570\u56e0\u6570<\/p>\n

\u4f18\u5316<\/h4>\n

\u4ec5\u9700\u679a\u4e3e\u5230\u8f83\u5c0f\u7684\u56e0\u5b50\u4e0a\u754c\u5373\u53ef\uff0c\u4e0a\u754c\u4e3a$\\scriptsize \\sqrt n$<\/p>\n

bool isPrime(int n) {\n    if(n <= 1) return false;\n    for(int i = 2; i * i <= n; i++)\n        if(n % i == 0) return false;\n    return true;\n}\n\nvector<int> enumeration(int l, int h) {\n    vector<int> v;\n    for(int i = l; i <= h; i++)\n        if(isPrime(i)) v.push_back(i);\n    return v;\n}\n<\/code><\/pre>\n
Eratosthenes-\u7b5b\u6cd5<\/h3>\n
\u533a\u95f4\u4e2d\u679a\u4e3e\u6240\u6709\u5408\u6570\uff0c\u7b5b\u6389\u6240\u6709\u5408\u6570\u5219\u5269\u4f59\u7684\u6570\u5747\u4e3a\u7d20\u6570<\/p>\n
\u6bcf\u4e2a\u5408\u6570\u5747\u53ef\u5206\u89e3\u4e3a \u6709\u9650\u4e2a\u8d28\u56e0\u5b50\u7684\u4e58\u79ef<\/strong>\uff0c\u4f9d\u6b21\u7b5b\u6389\u6240\u6709\u8d28\u56e0\u5b50\u7684\u500d\u6570\uff0c\u5219\u4ec5\u5269\u8d28\u56e0\u5b50\u672c\u8eab<\/p>\n
\u7b5b\u51fa\u533a\u95f4 $\\scriptsize [2,h]$ \u5185\u7684\u7d20\u6570\uff0c\u518d\u6311\u9009 $\\scriptsize [l,h]$ \u7684\u7d20\u6570<\/p>\n
\u8be6\u7ec6\u8bf4\u660e<\/h4>\n
\u4ece $\\scriptsize 2$ \u5f00\u59cb\u4e00\u8d9f\u7b5b\u53bb\u6240\u6709 $\\scriptsize 2N$ \uff0c\u4e00\u8d9f\u7b5b\u53bb\u6240\u6709 $\\scriptsize 3N$\uff0c\u4e00\u8d9f\u7b5b\u53bb\u6240\u6709 $\\scriptsize 5N$\uff0c… \uff0c\u4f9d\u6b21\u5411\u540e\u7b5b\uff0c\u5269\u4f59\u5e8f\u5217\u4e00\u5b9a\u4e3a\u7d20\u6570\u5e8f\u5217 $\\scriptsize 2,3,5,7,11,…$\uff0c\u7406\u7531\u5982\u4e0b<\/p>\n
\u533a\u95f4\u4e2d\u67d0\u6570 $\\scriptsize a$ \uff0c\u5728\u4e00\u8d9f\u533a\u95f4\u904d\u5386\u4e2d\u7b5b\u53bb $\\scriptsize [2,h]$ \u4e2d\u56e0\u5b50\u4e3a $\\scriptsize a$ \u7684\u6240\u6709\u5408\u6570 $\\scriptsize Na (N>1)$\uff0c\u4ee5 $\\scriptsize a$ \u4e3a\u56e0\u5b50\u7684\u6570\u4ec5\u5269 $\\scriptsize a$<\/p>\n
\u8bbe $\\scriptsize n \\subseteq N$\uff0c\u5219 $\\scriptsize na \\subseteq Na $\uff0c\u65e0\u9700\u518d\u7b5b $\\scriptsize na$ \u7684\u500d\u6570\uff0c\u56e0\u4e3a\u56e0\u5b50\u4e3a $\\scriptsize na$ \u7684\u5408\u6570\u5747\u5df2\u88ab\u7b5b<\/p>\n
\u5269\u4f59\u7684 $\\scriptsize a$ \u82e5\u4e3a\u5408\u6570\u5fc5\u5b9a\u5df2\u88ab\u4e4b\u524d\u7684\u56e0\u5b50\u7b5b\u6389\uff0c$\\scriptsize Na$ \u4e5f\u5fc5\u5b9a\u88ab\u7b5b\u6389\uff0c\u56e0\u6b64\u6b64\u65b9\u6cd5\u533a\u95f4\u5185\u5269\u4f59\u7684\u6570\u5747\u4e3a\u7d20\u6570<\/p>\n
\u6bcf\u4e2a\u5408\u6570\u8981\u88ab\u7b5b \u5408\u6570\u7684\u4e0d\u540c\u8d28\u56e0\u5b50\u4e2a\u6570<\/strong> \u6b21\uff0c\u5171\u9700\u7b5b $\\scriptsize n\/2 + n\/3 + n\/5 + … = n\\log \\log n$ \u6b21<\/p>\n
\u4f18\u5316<\/h4>\n
\u679a\u4e3e\u5408\u6570\u56e0\u5b50\u65f6\uff0c\u5230 $\\scriptsize \\sqrt n$ \u5373\u53ef\u679a\u4e3e\u51fa $\\scriptsize \\leqslant n$ \u7684\u6240\u6709\u5408\u6570<\/p>\n
\u679a\u4e3e\u5408\u6570\u65f6\uff0c\u540e\u7eed\u679a\u4e3e\u7684 $\\scriptsize (a+i)a$ \u5df2\u88ab $\\scriptsize a(a+i)$ \u679a\u4e3e\u8fc7\uff0c\u4e3a\u907f\u514d\u91cd\u590d\u4ece $\\scriptsize (a+i)^2, (a+i)(a+i+1), …$ \u7ee7\u7eed\u679a\u4e3e<\/p>\n
\u4f18\u5316\u540e\u4ecd\u6709\u91cd\u590d\uff0c\u5982 $\\scriptsize 2 \\times 6 = 3 \\times 4$<\/p>\n
const int maxn = 1000000;\nbool isPrime[maxn];\n\nvoid enumeration(int l, int h) {\n    memset(isPrime, true, sizeof(isPrime));\n\n    for (int i = 2; i * i <= h; i++)\n        if (isPrime[i])\n            for (int j = i * i; j <= h; j += i)\n                isPrime[j] = false;\n\n    for (int i = l; i <= h; i++)\n        if (isPrime[i])\n            printf(\"%d\", isPrime[i]);\n}\n<\/code><\/pre>\n
\u6b27\u62c9\u7ebf\u6027\u7b5b\u6cd5<\/h3>\n
E-\u7b5b\u6cd5\u4e2d\u6839\u636e\u5408\u6570\u7684\u6bcf\u4e2a\u8d28\u56e0\u5b50\u7b5b\u51fa\u5408\u6570\uff0c\u56e0\u800c\u5408\u6570\u91cd\u590d\u7b5b\u4e86\u8d28\u56e0\u5b50\u4e2a\u6570\u6b21\u3002<\/p>\n
\u6b27\u62c9\u7b5b\u6cd5\u4ec5 \u6839\u636e\u6700\u5c0f\u8d28\u56e0\u5b50\u7b5b\u51fa\u5408\u6570\uff0c\u56e0\u800c\u6bcf\u4e2a\u5408\u6570\u4ec5\u88ab\u7b5b\u4e00\u6b21<\/strong>\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u964d\u4e3a $\\scriptsize O(n)$\u3002<\/p>\n
\u8be6\u7ec6\u8bf4\u660e<\/h4>\n
\u4f9d\u6b21\u7b5b\u6389\u6240\u6709\u8d28\u56e0\u5b50\u7684\u500d\u6570\uff0c\u5219\u4ec5\u5269\u8d28\u56e0\u5b50\u672c\u8eab\uff0c\u4f9d\u6b21\u679a\u4e3e $\\scriptsize N\\times a(N \\in [2,n])$ \u5e76\u7b5b\u51fa\uff0c\u5176\u4e2d $\\scriptsize a$ \u4e3a\u5df2\u77e5\u8d28\u6570\uff0c\u5219\u5269\u4f59\u6570\u4e3a\u65b0\u786e\u5b9a\u7684\u5df2\u77e5\u8d28\u6570\uff0c\u7ee7\u7eed\u679a\u4e3e\u7b5b\u51fa\u3001\u7ee7\u7eed\u786e\u5b9a\uff0c…\uff0c\u5177\u4f53\u8fc7\u7a0b\u5982\u4e0b<\/p>\n\n\n\n\n\n\n\n\n\n\n
N<\/th>\n\u5df2\u77e5\u8d28\u6570<\/th>\n\u7b5b\u51fa\u5408\u6570<\/th>\n<\/tr>\n<\/thead>\n
2<\/td>\n2<\/td>\n4<\/td>\n<\/tr>\n
3<\/td>\n2, 3<\/td>\n6, 9<\/td>\n<\/tr>\n
4<\/td>\n2, 3<\/td>\n8<\/td>\n<\/tr>\n
5<\/td>\n2, 3, 5<\/td>\n10, 15, 25<\/td>\n<\/tr>\n
6<\/td>\n2, 3, 5<\/td>\n12<\/td>\n<\/tr>\n
7<\/td>\n2, 3, 5, 7<\/td>\n14, 21, 28, 35<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\u5bf9\u7167\u4e0a\u8868\uff0c\u4ee5\u4e0b\u89e3\u91ca\u6bcf\u4e2a\u5408\u6570\u4ec5\u88ab\u7b5b\u4e00\u6b21\u7684\u539f\u56e0\u3002\u5982\u679c\u679a\u4e3e $\\scriptsize 4 \\times a$ \u65f6\u53d6\u5230 $\\scriptsize a = 3$\uff0c\u679a\u4e3e $\\scriptsize 6 \\times a$ \u65f6\u53d6\u5230 $\\scriptsize a = 2$\uff0c\u5408\u6570 $\\scriptsize 12$ \u5c06\u88ab\u6700\u5c0f\u8d28\u56e0\u5b50 $\\scriptsize 2$ \u548c\u6b21\u5c0f\u8d28\u56e0\u5b50 $\\scriptsize 3$ \u5404\u4e58\u51fa\u4e00\u6b21\u5bfc\u81f4\u91cd\u590d\u3002\u8fd9\u662f\u56e0\u4e3a $\\scriptsize 4$ \u662f $\\scriptsize 12$ \u7684\u56e0\u5b50\uff0c\u6240\u4ee5 $\\scriptsize 12$ \u7684\u6700\u5c0f\u8d28\u56e0\u5b50\u5e94\u8be5\u662f $\\scriptsize 4$ \u7684\u6700\u5c0f\u8d28\u56e0\u5b50 $\\scriptsize 2$ \uff0c\u5176\u5bf9\u5e94\u6700\u5c0f\u8d28\u56e0\u5b50\u5206\u89e3\u4e3a $\\scriptsize 6 \\times 2$ \uff0c\u800c\u4e0d\u662f\u6b21\u5c0f\u8d28\u56e0\u5b50\u5206\u89e3 $\\scriptsize 4 \\times 3$<\/strong>\u3002\u56e0\u6b64\u679a\u4e3e\u56e0\u5b50 $\\scriptsize N=4$ \u53ea\u6709 $\\scriptsize 4 \\times 2 = 8$ \u53ef\u4fdd\u8bc1\u679a\u4e3e\u4e58\u79ef\u7684\u6700\u5c0f\u8d28\u56e0\u5b50\u4ecd\u4e3a  $\\scriptsize 2$\uff0c\u4e0e\u6240\u6709\u975e\u6700\u5c0f\u8d28\u56e0\u5b50\u7684\u4e58\u79ef\u90fd\u4e0d\u662f\u6700\u5c0f\u8d28\u56e0\u5b50\u5206\u89e3\uff0c\u90fd\u5c06\u4f1a\u5bfc\u81f4\u540e\u7eed\u91cd\u590d\u7b5b\u51fa\u3002<\/p>\n
\u56e0\u6b64\u679a\u4e3e\u7684\u56e0\u5b50 $\\scriptsize N$ \u4e3a\u5408\u6570\u65f6\u4ec5\u4e0e\u672c\u8eab\u7684\u6700\u5c0f\u8d28\u56e0\u5b50\u76f8\u4e58\uff0c\u679a\u4e3e\u7684\u56e0\u5b50\u4e3a\u8d28\u6570\u65f6\u4ec5\u4e0e\u4e0d\u5927\u4e8e\u672c\u8eab\u7684\u8d28\u56e0\u5b50\u76f8\u4e58\uff0c\u8fd9\u5c31\u4fdd\u8bc1\u4e86\u679a\u4e3e\u7684\u552f\u4e00\u6027<\/strong>\u3002\u56e0\u6b64\u5e94\u4f9d\u6b21\u6a21\u8fd0\u7b97\u68c0\u6d4b $\\scriptsize N$ \u662f\u5426\u5305\u542b\u8d28\u56e0\u5b50\u8868\u4e2d\u7684\u6700\u5c0f\u8d28\u56e0\u5b50\u3001\u6b21\u5c0f\u8d28\u56e0\u5b50\u3001…\uff0c\u4ee5\u4fdd\u8bc1\u6bcf\u4e2a\u679a\u4e3e\u51fa\u7684\u5408\u6570 $\\scriptsize Na$ \u90fd\u53ea\u88ab\u5176\u6700\u5c0f\u8d28\u56e0\u5b50\u7b5b\u51fa\uff0c\u90fd\u53ea\u88ab\u679a\u4e3e\u4e00\u6b21<\/p>\n
Q\uff1a<\/strong> \u65e2\u7136\u662f\u679a\u4e3e\u8d28\u56e0\u5b50\uff0c\u53ef\u5426\u4f18\u5316\u5230 $\\scriptsize O(\\sqrt n)$<\/p>\n
A\uff1a<\/strong> \u4e0d\u80fd\u3002\u679a\u4e3e\u7d20\u6570\u7b97\u6cd5\u4e0b\u754c\u662f\u533a\u95f4\u4e2d\u7d20\u6570\u7684\u4e2a\u6570 $\\scriptsize O(\\theta)$\uff0c\u8868\u793a\u6bcf\u4e2a\u7d20\u6570\u4ec5\u8ba1\u7b97\u4e00\u6b21\u5f97\u51fa\uff0c\u5176\u4e2d $\\scriptsize \\theta$ \u8fd1\u4f3c\u4e8e $\\scriptsize n\/\\ln n > \\sqrt n$\u3002\u5176\u6b21\uff1a\u2460 \u7ebf\u6027\u7b5b\u6cd5\u59cb\u7ec8\u6ee1\u8db3 $\\scriptsize N \\geqslant a$ \u2461 \u7ebf\u6027\u7b5b\u6cd5\u679a\u4e3e\u7684\u662f \u6700\u5c0f\u8d28\u56e0\u5b50\u7684\u5206\u89e3<\/strong> \u3002\u679a\u4e3e\u5230 $\\scriptsize N$ \u65f6\u57fa\u4e8e\u2460\u65e0\u6cd5\u53d6\u5230\u66f4\u5927\u7684\u8d28\u6570$\\scriptsize N’ (n > N’ > N)$\uff0c\u57fa\u4e8e\u2461\u533a\u95f4\u5185\u7684 $\\scriptsize N’a$ \u5747\u65e0\u6cd5\u88ab\u679a\u4e3e\u7b5b\u51fa\u3002\u5982\u679a\u4e3e $\\scriptsize [0, 100]$ \u7d20\u6570\uff0c\u5f53 $\\scriptsize N = 10$ \u65f6\uff0c$\\scriptsize 11, 13, 17, …$ \u7684\u500d\u6570\u5747\u672a\u88ab\u679a\u4e3e\u3002\u56e0\u6b64\uff0c\u8fbe\u5230 $\\scriptsize N = n\/2$ \u65f6\u624d\u53ef\u4fdd\u8bc1\u533a\u95f4\u6240\u6709\u5408\u6570\u7b5b\u5b8c\uff0c\u6b64\u65f6 $\\scriptsize isPrime$ \u5224\u522b\u8868\u586b\u5199\u5b8c\u6210\uff0c\u7b97\u6cd5\u5df2\u7ecf\u53ef\u4ee5\u505c\u6b62\u3002\u4e4b\u540e\u904d\u5386\u5230 $\\scriptsize n$ \u4ec5\u4ec5\u662f\u4e3a\u4e86\u586b\u5199\u5b8c\u6210\u7eaf\u7d20\u6570\u8868 $\\scriptsize a$\uff0c\u6b64\u8868\u7684\u7d22\u5f15 $\\scriptsize i$ \u5373\u662f\u7b2c $\\scriptsize i$ \u4e2a\u7d20\u6570\u7684\u5e8f\u53f7\uff0c\u4fbf\u4e8e\u4f9d\u636e\u5e8f\u53f7\u67e5\u627e\u7d20\u6570\uff1b\u82e5\u9700\u8981\u7eaf\u7d20\u6570\u8868\u65f6\uff0c\u4e5f\u65e0\u9700\u91cd\u65b0\u904d\u5386\u5224\u522b\u8868<\/p>\n
const int maxn = 1e6;\nbool isPrime[maxn];\nint a[maxn];\n\nvoid enumeration(int l, int h) {\n    int composite, cnt = 1;\n    memset(isPrime, true, sizeof(isPrime));\n\n    for (int N = 2; N <= h; N++) {\n        if (isPrime[N]) a[cnt++] = N;\n        for (int i = 1; i < cnt; i++) {\n            if ((composite = N * a[i]) > h) break;\n            isPrime[composite] = false;\n            if (N % a[i] == 0) break;\n        }\n    }\n\n    for (int i = l; i <= h; i++)\n        if (isPrime[i])\n            printf(\"%d\", isPrime[i]);\n}\n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
\u7efc\u8ff0 \u679a\u4e3e\u533a\u95f4 $\\scriptsize [l, h]$ \u5185\u7684\u6240\u6709\u7d20\u6570\u95ee\u9898\uff0c\u65b9\u6cd5\u53ca\u65f6\u95f4\u590d\u6742\u5ea6\u5982\u4e0b \u65b9\u6cd5 \u601d\u60f3  … <\/p>\n
\u7ee7\u7eed\u9605\u8bfb\u201c\u7ebf\u6027\u65f6\u95f4\u679a\u4e3e\u7d20\u6570 – \u6570\u8bba\u65b9\u6cd5\u201d<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[12,1],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts\/1672"}],"collection":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1672"}],"version-history":[{"count":57,"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts\/1672\/revisions"}],"predecessor-version":[{"id":2031,"href":"http:\/\/139.196.114.170\/index.php?rest_route=\/wp\/v2\/posts\/1672\/revisions\/2031"}],"wp:attachment":[{"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1672"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1672"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/139.196.114.170\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1672"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}