ACM-ICPC 2018 南京赛区网络预赛-J_sum
A square-free integer is an integer which is indivisible by any square number except 11. For example, 6 = 2 \cdot 36=2⋅3 is square-free, but 12 = 2^2 \cdot 312=22⋅3 is not, because 2^222 is a square number. Some integers could be decomposed into product of two square-free integers, there may be more than one decomposition ways. For example, 6 = 1\cdot 6=6 \cdot 1=2\cdot 3=3\cdot 2, n=ab6=1⋅6=6⋅1=2⋅3=3⋅2,n=ab and n=ban=ba are considered different if a \not = ba̸=b. f(n)f(n) is the number of decomposition ways that n=abn=ab such that aa and bb are square-free integers. The problem is calculating \sum_{i = 1}^nf(i)∑i=1nf(i).
Input
The first line contains an integer T(T\le 20)T(T≤20), denoting the number of test cases.
For each test case, there first line has a integer n(n \le 2\cdot 10^7)n(n≤2⋅107).
Output
For each test case, print the answer \sum_{i = 1}^n f(i)∑i=1nf(i).
Hint
\sum_{i = 1}^8 f(i)=f(1)+ \cdots +f(8)∑i=18f(i)=f(1)+⋯+f(8)
=1+2+2+1+2+4+2+0=14=1+2+2+1+2+4+2+0=14.
样例输入复制
2
5
8样例输出复制
8
14
唉，要是想到分块，早就Ac了，现实却是到比赛结束也没想到。。。o(╥﹏╥)o
当时暴力把所有数据打出来了（200秒）。。每一个F(x)（不是ans(x)），一共40多MB。。。
思路
将每一个数唯一分解
设X = (k1^n1)*(K2^n2)*(K3^n3)...........
那么将X分解为a*b的形式，只要选出a就行了，剩下的就是b
那么，
如果ni等于1，那么可以选，可以不选
如果等于2，一定需要选取（相当于没有这个Ki），
---------------这里说一下吧例如（2^2）*3*5，为了分离出符合条件的a，b，
必须是a中选一个2，b中选一个2，否则a或b中一定有个2的平方，因为2已经确定是一定要选的了，
那么只需要在剩下的3和5中选择就可以了，二项式基本定理（大概是叫这个来着）得出2的2次方种取法
大于2，F(X)=0，当前X不能分解出有效的a*b
那么就是计算选取的方式有多少种
把ni等于2的看成没有这个ki，那么就是问有多少个ni等于1
可选可不选就是2^count(ni==1)种可能
那么暴力求解即可
-------------------------分块的思想
如果要求1-----100内的数，而求直接求超时的话可以考虑每隔10（100开根号）个数记录一下ans，
那么每次最多就跑sqrt(n)（假设k不在记录点，那么k一定在两个记录点之间，而记录点间隔是sqrt（n）），
这样硬生生的优化到了sqrt(n)，暴力真好用，下次我还考虑它
注：sqrt（2*1e7）=4472
对了，还有个小技巧：就是唯一分解或者说求一个数的因子的时候，只要求到sqrt（n）就行了，（类似素数筛选）
连续除掉因子，这样n的大小不断缩小，效率也变快了
但是有可能这样，
例如某个大素数*2之类的，这样就不能简单的到sqrt（n），最后要判断一下是否大于1，这样说明还有个大素数
#include
using namespace std;
typedef long long ll;
int vis[10005];
int prim[2000];
int Pr_cnt,Pr_mn = 10000;
int Fx[4500]={
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532417,138565867,138599262,138632796,138666250,138699750,138733202,138766584,138799924,138833330,138866831,138900371,138933864,138967012,139000453,139034009,139067435,139100987,139134293,139167823,139201216,139234812,139268151,139301503,139334857,139368185,139401666,139435150,139468693,139502207,139535626,139568950,139602392,139635664,139669149,139702719,139736004,139769480,139802817,139836405,139869841,139903203,139936635,139969921,140003658,140036856,140070379,140103743,140137213,140170717,140204140,140237544,140270879,140304309,140337868,140371280,140404748,140438160,140471409,140505033,140538319,140571935,140605406,140638830,140672378,140705632,140739099,140772633,140806077,140839591,140872938,140906400,140939786,140973508,141006825,141040427,141073696,141107062,141140480,141173798,141207286,141240904,141274289,141307683,141341276,141374704,141408284,141441792,141475106,141508684,141541983,141575581,141609024,141642328,141675857,141709389,141742823,141776261,141809778,141843232,141876611,141910089,141943432,141976838,142010312,142043830,142077381,142110665,142144098,142177754,142211175,142244603
};///4472
void Eulerprim()
{
for(int i=2;i<=Pr_mn;i++)
{
if(!vis[i])
prim[++Pr_cnt] = i;
for(int j=1;j<=Pr_cnt&&prim[j]*i<=Pr_mn;j++)
{
vis[prim[j]*i] = 1;
if(i%prim[j]==0)
break;
}
}
}
int init(int Begin,int End)
{
int ans = 0;
for(int i=Begin;i<=End;i++)
{
int ti = i;
int cnt = 0,flag = 0;
for(int j=1;jti)break;
if(ti%prim[j]==0)
{
cnt++;
int c = 0;
while(ti%prim[j]==0)
{
ti/=prim[j],c++;
if(c>2)
{
flag = 1;
break;
}
}
if(flag)
break;
if(c==2)
cnt--;
}
}
if(ti>1)
cnt++;
if(!flag)
ans += 1< | 正确通过 | 2018-09-01 20:40 | 108ms | 284kB | c++ |
| 正确通过 | 2018-09-01 20:03 | 107ms | 284kB | c++ |
写在最后的话
这只是个笨办法，要是能得到更简单的方法，还是不要考虑这个了
搜了一下，找了几个用其他方法的（都比我要强。。。）
欧拉筛选法，可怜我筛选素数用的也是欧拉。。传送门
还有dp传送门
这个也是筛选，但不是欧拉传送门