【牛客想开了大赛2 B】n的约数【打表】

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博客围绕一个算法问题展开，题目为t次询问，每次给定数n，求[1,n]内约数个数最多的数的约数个数。解题思路是打表，此为经典反素数问题，在1018范围内符合条件的数字仅156个，可通过OEIS找表，还给出了具体操作步骤。

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题目大意：

题目链接：https://ac.nowcoder.com/acm/contest/907/B
$t$ 次询问，每次给你一个数 $n$ ，求在 $[1, n]$ 内约数个数最多的数的约数个数。

思路：

打表。
在 $[1, n]$ 内约数个数最多的数就是经典的反素数问题。
通过OEIS发现，这样的数字在 $10^{18}$ 范围内只有156个。

然后OEIS都是会有表的
在这里插入图片描述
而这156个数字就是小于等于 $10^{18}$ 的约数个数最多的数字。

string a[158]={"0","1","2","4","6","12","24","36","48","60","120","180","240","360","720","840","1260","1680","2520","5040","7560","10080","15120","20160","25200","27720","45360","50400","55440","83160","110880","166320","221760","277200","332640","498960","554400","665280","720720","1081080","1441440","2162160","2882880","3603600","4324320","6486480","7207200","8648640","10810800","14414400","17297280","21621600","32432400","36756720","43243200","61261200","73513440","110270160","122522400","147026880","183783600","245044800","294053760","367567200","551350800","698377680","735134400","1102701600","1396755360","2095133040","2205403200","2327925600","2793510720","3491888400","4655851200","5587021440","6983776800","10475665200","13967553600","20951330400","27935107200","41902660800","48886437600","64250746560","73329656400","80313433200","97772875200","128501493120","146659312800","160626866400","240940299600","293318625600","321253732800","481880599200","642507465600","963761198400","1124388064800","1606268664000","1686582097200","1927522396800","2248776129600","3212537328000","3373164194400","4497552259200","6746328388800","8995104518400","9316358251200","13492656777600","18632716502400","26985313555200","27949074753600","32607253879200","46581791256000","48910880818800","55898149507200","65214507758400","93163582512000","97821761637600","130429015516800","195643523275200","260858031033600","288807105787200","391287046550400","577614211574400","782574093100800","866421317361600","1010824870255200","1444035528936000","1516237305382800","1732842634723200","2021649740510400","2888071057872000","3032474610765600","4043299481020800","6064949221531200","8086598962041600","10108248702552000","12129898443062400","18194847664593600","20216497405104000","24259796886124800","30324746107656000","36389695329187200","48519593772249600","60649492215312000","72779390658374400","74801040398884800","106858629141264000","112201560598327200","149602080797769600","224403121196654400","299204161595539200","374005201994424000","448806242393308800","673209363589963200","748010403988848000","897612484786617600","999999999999999999999999"};

然后把前面几个最小的数字拿出来，暴力算出它的约数个数。

#include <cstdio>
#include <iostream>

int cnt,n;

int main()
{
	scanf("%d",&n);
	for (int i=1;i<=n;i++)
		if (!(n%i)) cnt++;
	printf("%d\n",cnt);
	return 0;
}

然后把这些数字再次放到OEIS上
找到表的前156个数字

string b[157]={"0","1","2","3","4","6","8","9","10","12","16","18","20","24","30","32","36","40","48","60","64","72","80","84","90","96","100","108","120","128","144","160","168","180","192","200","216","224","240","256","288","320","336","360","384","400","432","448","480","504","512","576","600","640","672","720","768","800","864","896","960","1008","1024","1152","1200","1280","1344","1440","1536","1600","1680","1728","1792","1920","2016","2048","2304","2400","2688","2880","3072","3360","3456","3584","3600","3840","4032","4096","4320","4608","4800","5040","5376","5760","6144","6720","6912","7168","7200","7680","8064","8192","8640","9216","10080","10368","10752","11520","12288","12960","13440","13824","14336","14400","15360","16128","16384","17280","18432","20160","20736","21504","23040","24576","25920","26880","27648","28672","28800","30720","32256","32768","34560","36864","40320","41472","43008","46080","48384","49152","51840","53760","55296","57600","61440","62208","64512","65536","69120","73728","80640","82944","86016","92160","96768","98304","103680"};

然后就可以暴力求了。

代码:

#include <string>
#include <iostream>
using namespace std;

string a[158]={"0","1","2","4","6","12","24","36","48","60","120","180","240","360","720","840","1260","1680","2520","5040","7560","10080","15120","20160","25200","27720","45360","50400","55440","83160","110880","166320","221760","277200","332640","498960","554400","665280","720720","1081080","1441440","2162160","2882880","3603600","4324320","6486480","7207200","8648640","10810800","14414400","17297280","21621600","32432400","36756720","43243200","61261200","73513440","110270160","122522400","147026880","183783600","245044800","294053760","367567200","551350800","698377680","735134400","1102701600","1396755360","2095133040","2205403200","2327925600","2793510720","3491888400","4655851200","5587021440","6983776800","10475665200","13967553600","20951330400","27935107200","41902660800","48886437600","64250746560","73329656400","80313433200","97772875200","128501493120","146659312800","160626866400","240940299600","293318625600","321253732800","481880599200","642507465600","963761198400","1124388064800","1606268664000","1686582097200","1927522396800","2248776129600","3212537328000","3373164194400","4497552259200","6746328388800","8995104518400","9316358251200","13492656777600","18632716502400","26985313555200","27949074753600","32607253879200","46581791256000","48910880818800","55898149507200","65214507758400","93163582512000","97821761637600","130429015516800","195643523275200","260858031033600","288807105787200","391287046550400","577614211574400","782574093100800","866421317361600","1010824870255200","1444035528936000","1516237305382800","1732842634723200","2021649740510400","2888071057872000","3032474610765600","4043299481020800","6064949221531200","8086598962041600","10108248702552000","12129898443062400","18194847664593600","20216497405104000","24259796886124800","30324746107656000","36389695329187200","48519593772249600","60649492215312000","72779390658374400","74801040398884800","106858629141264000","112201560598327200","149602080797769600","224403121196654400","299204161595539200","374005201994424000","448806242393308800","673209363589963200","748010403988848000","897612484786617600","999999999999999999999999"};        
string b[157]={"0","1","2","3","4","6","8","9","10","12","16","18","20","24","30","32","36","40","48","60","64","72","80","84","90","96","100","108","120","128","144","160","168","180","192","200","216","224","240","256","288","320","336","360","384","400","432","448","480","504","512","576","600","640","672","720","768","800","864","896","960","1008","1024","1152","1200","1280","1344","1440","1536","1600","1680","1728","1792","1920","2016","2048","2304","2400","2688","2880","3072","3360","3456","3584","3600","3840","4032","4096","4320","4608","4800","5040","5376","5760","6144","6720","6912","7168","7200","7680","8064","8192","8640","9216","10080","10368","10752","11520","12288","12960","13440","13824","14336","14400","15360","16128","16384","17280","18432","20160","20736","21504","23040","24576","25920","26880","27648","28672","28800","30720","32256","32768","34560","36864","40320","41472","43008","46080","48384","49152","51840","53760","55296","57600","61440","62208","64512","65536","69120","73728","80640","82944","86016","92160","96768","98304","103680"};
string s;
int t;

bool check(string s1,string s2)
{
	if (s1.size()<s2.size()) return 1;
	if (s1.size()>s2.size()) return 0;
	for (int i=0;i<s1.size();i++)
	{
		if (s1[i]>s2[i]) return 0;
		if (s1[i]<s2[i]) return 1;
	}
	return 0;
}

int main()
{
	cin>>t;
	while (t--)
	{
		cin>>s;
		for (int i=1;i<=157;i++)
			if (check(s,a[i]))
			{
				cout<<b[i-1]<<endl;
				break;
			}
	}
	return 0;
}