「FCS NOI 2018 day1」最大真因数 - 分段打表 | Bill Yang's Blog

「FCS NOI 2018 day1」最大真因数 - 分段打表

题目大意

    一个合数的真因数是指这个数不包括其本身的所有因数,例如$6$的正因数有$1,2,3,6$,其中真因数有$1,2,3$。一个合数的最大真因数则是这个数的所有真因数中最大的一个,例如$6$的最大真因数为$3$。
    给定正整数$l$和$r$,请你求出$l$和$r$之间(包括$l$和$r$)所有合数的最大真因数之和。


题目分析

这不是标解,标解是埃筛。

最大真因数$=$原数$/$最小质因数。
因此求出最小质因数即可。
对于$r-l\le5\times10^6$的情况,显然筛出一遍素数后枚举倍数即可,时间复杂度$O(\frac{n}{\ln n}\times\log n)=O(n)$。
对于其他情况,我们可以分段打表。
首先预处理出$1\sim5\times10^6,5\times10^6+1\sim1\times10^7,\ldots,5\times10^9-5\times10^6+1\sim5\times10^9$的真因数之和,打表存下来。
接下来将询问区间边界的不完整区间用暴力计算,剩下$5\times10^6$的整块,利用打好的表取个和即可。


代码

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#include<bits/stdc++.h>

using namespace std;

typedef long long LL;

inline LL Get_Int() {
LL num=0,bj=1;
char x=getchar();
while(!isdigit(x)) {if(x=='-')bj=-1;x=getchar();}
while(isdigit(x)) {num=num*10+x-'0';x=getchar();}
return num*bj;
}

const int maxv=5000000,maxn=maxv+5;

LL l,r;
int pr[maxn],Min[maxn],cnt=0;
bool vst[maxn];

void Prime_Table(int x) {
for(int i=2; i<=x; i++) {
if(!vst[i])pr[++cnt]=i;
for(int j=1; j<=cnt&&i*pr[j]<=x; j++) {
vst[i*pr[j]]=1;
if(i%pr[j]==0)break;
}
}
}

namespace Solve1 {
void solve() {
int delta=r-l;
LL ans=0;
for(int i=1; i<=cnt; i++)
for(LL j=max(2ll,(l-1)/pr[i]+1)*pr[i]; j<=r; j+=pr[i])
if(!Min[j-l])Min[j-l]=pr[i];
for(int i=0; i<=delta; i++)if(Min[i])ans+=(i+l)/Min[i];
printf("%lld\n",ans);
}
}

namespace Solve2 {
LL f[]= 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LL ans=0;
void solve(LL l,LL r) {
int delta=r-l;
for(int i=1; i<=cnt; i++)
for(LL j=max(2ll,(l-1)/pr[i]+1)*pr[i]; j<=r; j+=pr[i])
if(!Min[j-l])Min[j-l]=pr[i];
for(int i=0; i<=delta; i++)if(Min[i])ans+=(i+l)/Min[i];
}
void solve() {
LL l1=l,r1=((l-1)/maxv+1)*maxv;
solve(l1,r1);
memset(Min,0,sizeof(Min));
LL l2=(r-1)/maxv*maxv+1,r2=r;
solve(l2,r2);
int l3=r1/maxv+1,r3=(l2-1)/maxv;
for(int i=l3; i<=r3; i++)ans+=f[i];
printf("%lld\n",ans);
}
}

int main() {
l=Get_Int();
r=Get_Int();
Prime_Table(sqrt(r)+5);
if(r-l<=maxv)Solve1::solve();
else Solve2::solve();
fclose(stdin);
fclose(stdout);
return 0;
}
姥爷们赏瓶冰阔落吧~
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