思考歪了.

原本想的是,设1的长度为n,有

式子左半部分是999…999(有n/2个,因为他们数位和一定是9的倍数,所以左式一定能被9整除,变成111…111,同样是n/2个),

上面的方法不能解决奇数n的问题,本题因此卡住.

其实应该从取模的意义考虑的,假设

放大限制: $p|{10^n-1}$ ,直接判断即可.

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#include<bits/stdc++.h>
using namespace std;
#define enter fout<<"\n";
#define space fout<<" ";
#define dot fout<<",";
#define oui fout<<"Yes\n";
#define non fout<<"No\n";
#define si fout<<"?";
#define i32 int
#define u32 unsigned int
#define i64 long long
#define u64 unsigned long long
#define i128 __int128
#define u128 unsigned __int128
#define debug(x) fout<<#x<<"="<<x<<"\n";
#define vdebug(a,n) fout<<#a<<"=[";for(int i=0;i<=n;++i)fout<<a[i]<<" ";fout<<"]\n";
mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
namespace fastio{
    const int bufl=1<<20;
    const double base1[16]={1,1e-1,1e-2,1e-3,1e-4,1e-5,1e-6,1e-7,1e-8,1e-9,1e-10,1e-11,1e-12,1e-13,1e-14,1e-15};
    const double base2[16]={1,1e1,1e2,1e3,1e4,1e5,1e6,1e7,1e8,1e9,1e10,1e11,1e12,1e13,1e14,1e15};
    struct IN{
        FILE *IT;char ibuf[bufl],*is=ibuf,*it=ibuf;
        IN(){IT=stdin;}IN(char *a){IT=fopen(a,"r");}
        inline char getChar(){if(is==it){it=(is=ibuf)+fread(ibuf,1,bufl,IT);if(is==it)return EOF;}return *is++;}
        template<typename Temp>inline void getInt(Temp &a){a=0;int b=0,c=getChar();while(c<48||c>57)b^=(c==45),c=getChar();while(c>=48&&c<=57)a=(a<<1)+(a<<3)+c-48,c=getChar();if(b)a=-a;}
        template<typename Temp>inline void getDouble(Temp &a){a=0;int b=0,c=getChar(),d=0;__int128 e=0,f=0;while(c<48||c>57)b^=(c==45),c=getChar();while(c>=48&&c<=57)e=(e<<1)+(e<<3)+c-48,c=getChar();if(c==46){c=getChar();while(c>=48&&c<=57)d++,f=(f<<1)+(f<<3)+c-48,c=getChar();}a=e+base1[d]*f;if(b)a=-a;}
        IN& operator>>(char &a){a=getChar();while(a<=32)a=getChar();return *this;}
        IN& operator>>(char *a){do{*a=getChar();}while(*a<=32);while(*a>32)*++a=getChar();*a=0;return *this;}
        IN& operator>>(string &a){a.clear();char b=getChar();while(b<=32)b=getChar();while(b>32)a+=b,b=getChar();return *this;}
        IN& operator>>(int &a){getInt(a);return *this;}
        IN& operator>>(long long &a){getInt(a);return *this;}
        IN& operator>>(__int128 &a){getInt(a);return *this;}
        IN& operator>>(float &a){getDouble(a);return *this;}
        IN& operator>>(double &a){getDouble(a);return *this;}
        IN& operator>>(long double &a){getDouble(a);return *this;}
    };
    struct OUT{
        FILE *IT;char obuf[bufl],*os=obuf,*ot=obuf+bufl;int Eps;long double Acc;
        OUT(){IT=stdout,Eps=6,Acc=0.5;}OUT(char *a){IT=fopen(a,"w"),Eps=6,Acc=0.5;}
        inline void ChangEps(int x=6){Eps=x;}
        inline void flush(){fwrite(obuf,1,os-obuf,IT);os=obuf;}
        inline void putChar(int a){*os++=a;if(os==ot)flush();}
        template<typename Temp>inline void putInt(Temp a){if(a<0){putChar(45);a=-a;}if(a<10){putChar(a+48);return;}putInt(a/10);putChar(a%10+48);}
        template<typename Temp>inline void putLeading(Temp a,int b){if(!b)return;putLeading(a/10,b-1);putChar(a%10+48);}
        template<typename Temp>inline void putDouble(Temp a){if(a<0){putChar(45);a=-a;}__int128 ff=(a-(__int128)a)*base2[Eps+2],gg=0;ff+=50;while(ff>0){ff/=10;gg++;}__int128 b=a;if(gg==Eps+3){putInt(b+1);}else{putInt(b);}a-=b;a*=base2[Eps];b=a+Acc;putChar(46);putLeading(b,Eps);}
        OUT& operator<<(char a){putChar(a);return *this;}
        OUT& operator<<(const char *a){while(*a)putChar(*a++);return *this;}
        OUT& operator<<(string a){for(auto c:a)putChar(c);return *this;}
        OUT& operator<<(int a){putInt(a);return *this;}
        OUT& operator<<(long long a){putInt(a);return *this;}
        OUT& operator<<(__int128 a){putInt(a);return *this;}
        OUT& operator<<(unsigned int a){putInt(a);return *this;}
        OUT& operator<<(unsigned long long a){putInt(a);return *this;}
        OUT& operator<<(unsigned __int128 a){putInt(a);return *this;}
        OUT& operator<<(float a){putDouble(a);return *this;}
        OUT& operator<<(double a){putDouble(a);return *this;}
        OUT& operator<<(long double a){putDouble(a);return *this;}
        ~OUT(){flush();}
    };
}
fastio::IN fin;
fastio::OUT fout;

template<typename T1,typename T2,typename T3>
T1 qp(T1 b,T2 po,T3 p){
    T1 res(1);
    while(po>0){
        if(po&1)
            res=res*b%p;
        b=b*b%p;
        po>>=1;
    }
    return res;
}
int   vis[1000010];//存最小质因数,负的表示质数表中的位置(负的)
int  p[100010],ptop=0;//存质数 
short  mu[1000010];//莫比乌斯函数 
int  musu[1000010];//梅滕斯函数,莫比乌斯前缀和 
int   phi[1000010];//欧拉函数 
long long phisu[1000010];//欧拉函数前缀和 
int	 d[1000010];//存每个数的约数个数 
int mnnum[1000010];//最小质因子出现次数 
void sieve(int n){//[1,n]
    phi[1]=1;phisu[1]=1;mu[1]=1;musu[1]=1;d[1]=1;
    int tmp;
    for(int i=2;i<=n;++i){
        if(!vis[i]){
            vis[i]=-(++ptop);
            p[ptop]=i;
            mu[i]=-1;//
            phi[i]=i-1;//
            d[i]=2;//
            mnnum[i]=1;//
        }
        for(int j=1;j<=ptop&&i*p[j]<=n;++j){
            vis[i*p[j]]=p[j];
            if(i%p[j]==0){
                phi[i*p[j]]=phi[i]*p[j];//
                mnnum[i*p[j]]=mnnum[i]+1;//
                d[i*p[j]]=d[i]/mnnum[i*p[j]]*(mnnum[i*p[j]]+1);//
                break;
            }else{
                mu[i*p[j]]=-mu[i];//
                phi[i*p[j]]=phi[i]*(p[j]-1);//
                mnnum[i*p[j]]=1;//
                d[i*p[j]]=d[i]*2;//
            }
        }
        musu[i]=musu[i-1]+mu[i];//
        phisu[i]=phisu[i-1]+phi[i];//
    }
}





char chk(int p){
    i64 num=qp((i64)10,1000000000%(p-1),p);
    // num*=qp((i64)9,p-2,p);
    // num%=p;
    if(num==1)return 1;
    else return 0;
}

long long n,m,res;
//#define NaraFluorine
int main(){
    sieve(1000000);
    int cnt=0;
    for(int i=4;i<=ptop;++i){
        if(cnt>=40)break;
        if(chk(p[i])){
            cnt++;
            res+=p[i];
            fout<<cnt<<" "<<p[i]<<" "<<res<<"\n";
        }
    }
    return 0;
}