数论板子(持续更新

bool issqr(__int128_t x)//开方
{
__int128_t y=(__int128_t)ceil(sqrt((long double)x));
for(;y*y<=x;++y);
for(--y;y*y>x;--y);
return y*y==x;
}
bool iscub(__int128_t x)//三次方
{
__int128_t y=(__int128_t)ceil(pow((long double)x,1.0/3));
for(;y*y*y<=x;++y);
for(--y;y*y*y>x;--y);
return y*y*y==x;
}

int tot,pr[maxx],vis[maxx];
void init()//素数表  O(nsqrt(n))
{
for(int i=2;i<maxx;i++)
{
if(!vis[i])pr[tot++]=i;
for(int j=0;j<tot&&i*pr[j]<maxx;j++)
{
vis[i*pr[j]]=1;
if(i%pr[j]==0)break;
}
}
} 

int d[maxx],pr[maxx],tot;
void init()
{
    for(int i = 2; i < maxx; ++i)
    {
        if(!d[i])
            pr[tot++] = d[i] = i;
        for(int j = 0, k; (k = i * pr[j]) < maxx; ++j)
        {
            d[k] = pr[j];
            if(d[i] == pr[j])
                break;
        }
    }
}

const int s=10;//Miller-Rabin
inline LL mul_mod(LL a, LL b, LL m)  O(n^(1/4))
{
LL c = a*b-(LL)((long double)a*b/m+0.5)*m;
return c<0 ? c+m : c;
}

LL fast_exp(LL a,LL x,LL m)
{
LL b = 1;
while (x)
{
if (x & 1)
b = mul_mod(b, a, m);
a = mul_mod(a, a, m);
x >>= 1;
}
return b;
}

bool MR(LL n)
{
if (!(n&1))
return n == 2;
LL t = 0, u;
for (u = n-1; !(u&1); u >>= 1)
++t;
for (int i = 0; i < s; ++i)
{
LL a = rand()%(n-2)+2, x = fast_exp(a, u, n);
for (LL j = 0, y; x != 1 && j < t; ++j, x = y)
{
y = mul_mod(x, x, n);
if (y == 1 && x != n-1)
return false;
}
if (x != 1)
return false;
}
return true;
}

Miller-Rabin Pollard-rho

#include <cstdio>
#include <cstdlib>
#include <ctime>
#include <algorithm>
typedef long long ll;
const int s = 10, MAX_F = 70;
ll cnt, f[MAX_F];

inline ll mul_mod(ll a, ll b, ll m)
{
ll c = a*b-(ll)((long double)a*b/m+0.5)*m;
return c<0 ? c+m : c;
}

ll fast_exp(ll a, ll x, ll m)
{
ll b = 1;
while (x)
{
if (x & 1)
b = mul_mod(b, a, m);
a = mul_mod(a, a, m);
x >>= 1;
}
return b;
}

bool MR(ll n)
{
if (!(n&1))
return n == 2;
ll t = 0, u;
for (u = n-1; !(u&1); u >>= 1)
++t;
for (int i = 0; i < s; ++i)
{
ll a = rand()%(n-2)+2, x = fast_exp(a, u, n);
for (ll j = 0, y; x != 1 && j < t; ++j, x = y)
{
y = mul_mod(x, x, n);
if (y == 1 && x != n-1)
return false;
}
if (x != 1)
return false;
}
return true;
}

inline ll abs(ll x)
{
return x<0 ? -x : x;
}

ll gcd(ll a, ll b)
{
return b ? gcd(b, a%b) : a;
}

ll PR(ll n, ll a)
{
ll x = rand()%n, y = x, k = 1, i = 0, d = 1;
while (d == 1)
{
if ((x = (mul_mod(x, x, n)+a)%n) == y)
return n;
d = gcd(n, abs(y-x));
if (++i == k)
{
k <<= 1;
y = x;
}
}
return d;
}

void decomp(ll n)
{
if (n == 1)
return;
if (MR(n))
{
f[cnt++] = n;
return;
}
ll d = n, c = n-1;
while (d == n)
d = PR(n, c--);
do
{
n /= d;
}
while (!(n%d));
decomp(d);
decomp(n);
}

int main()
{
srand(time(0));
ll n;
while (scanf("%lld", &n), n)
{
cnt = 0;
decomp(n);
std::sort(f, f+cnt);
cnt = std::unique(f, f+cnt)-f;
ll ans = n;
for (int i = 0; i < cnt; ++i)
{
// printf("%lld\n", f[i]);
ans = ans/f[i]*(f[i]-1);
}
printf("%lld\n", ans);
}
return 0;
}

快速乘

LL fun(LL a,LL b)
{
LL sum=0;
W(b)
{
if(b&1)
sum=(sum+a)%p;
b/=2;
a=a*2%p;
}
return sum;
}

快速幂

int _pow(LL a,LL n)
{
LL ret=1;
while(n)
{
if(n&1)  ret=ret*a%mod;
a=a*a%mod;
n>>=1;
}
return ret;
}
int inv(int x)
{
return _pow(x,mod-2);
}

拓展gcd

void exgcd(ll a,ll b,ll &g,ll &x,ll &y)
{
if(!b) {g=a;x=1;y=0;}
else {exgcd(b,a%b,g,y,x);y-=x*(a/b);}
}
void work(ll a , ll b , ll d ,ll& g , ll& x , ll& y)
{
exgcd(a,b,g,x,y);      //此处的x，y接收 ax+by=gcd(a,b)的值
x *= d/g;             //求ax+by=c 的解x
int t = b/g;
x = (x%t + t) % t;
y = abs( (a*x - d) / b);
}
ll a,b,g,x,y,d;
void solve()
{
S_3(a,b,d);
work(a,b,d,g,x,y);
/* if(x==0)
{
for(int i=1;x<=0;i++)
{
x=x+b/g*i;
y=y-a/g*i;
}
}*/
}

逆元

void ex_gcd(LL a, LL b, LL &x, LL &y, LL &d)
{
if (!b) {d = a, x = 1, y = 0;}
else{
ex_gcd(b, a % b, y, x, d);
y -= x * (a / b);
}
}
LL inv2(LL t, LL p)
{//如果不存在，返回-1
LL d, x, y;
ex_gcd(t, p, x, y, d);
return d == 1 ? (x % p + p) % p : -1;
}

错排 排列组合

const int mod=1e9+7;
const int maxn=1e4+7;
LL D[maxn], C[maxn][105];
int p[1000050];
void init()
{
D[0]=1;D[1]=0;
for(int i=1;i<maxn;i++)//错排
D[i]=((i-1)*(D[i-1]+D[i-2])+mod)%mod;
int i;
for (p[0]=i=1;i<=200000;i++) p[i]=1ll*p[i-1]*i%mod;
for (int i = 0; i < maxn; i++)//排列组合
{
for (int j = 0; j <= min(105, i); j++)
{
if (j == 0) C[i][j] = 1;
else
C[i][j] = (C[i-1][j] + C[i-1][j-1]) % mod;
}
}
}

C(n,m)

先init 再comb(n,m) n在下

namespace COMB
{
int F[N<<1], Finv[N<<1], inv[N<<1];
void init()
{
inv[1] = 1;
for (int i = 2; i < N<<1; i++)
{
inv[i] = (LL)(MOD - MOD/i) * inv[MOD%i] % MOD;
}
F[0] = Finv[0] = 1;
for (int i = 1; i < N<<1; i++)
{
F[i] = (LL)F[i-1] * i % MOD;
Finv[i] = (LL)Finv[i-1] * inv[i] % MOD;
}
}
int comb(int n, int m)
{
if (m < 0 || m > n) return 0;
return (LL)F
* Finv[n-m] % MOD * Finv[m] % MOD;
}
}
using namespace COMB;

C（n,m） 不mod

double ln[3100000];
int n,p;
void init()
{
int maxc=3e6;
for(int i=1;i<=maxc;i++)
{
ln[i]=log(i);
}
}

db comb(int x,int y)
{
db ans=0;
for(int i=1;i<=y;i++)
{
ans+=ln[x-i+1]-ln[i];
}
return ans;
}

db calc(int x)
{
return x*exp(comb(n,p-1)-comb(n+x,p));
}

C(n,m) mod p (p为素数

//C(m,n) m在上 n在下,p为素数
LL n,m,p;

LL quick_mod(LL a, LL b)
{
LL ans = 1;
a %= p;
while(b)
{
if(b & 1)
{
ans = ans * a % p;
b--;
}
b >>= 1;
a = a * a % p;
}
return ans;
}

LL C(LL n, LL m)
{
if(m > n) return 0;
LL ans = 1;
for(int i=1; i<=m; i++)
{
LL a = (n + i - m) % p;
LL b = i % p;
ans = ans * (a * quick_mod(b, p-2) % p) % p;
}
return ans;
}

LL Lucas(LL n, LL m)
{
if(m == 0) return 1;
return C(n % p, m % p) * Lucas(n / p, m / p) % p;
}

int main()
{
int T;
scan_d(T);
while(T--)
{
scan_d(n),scan_d(m),scan_d(p);
print(Lucas(n,m));
}
return 0;
}

C(n,m) mod p (p可能为合数

const int N = 200005;

bool prime
;
int p
;
int cnt;

void isprime()
{
cnt = 0;
memset(prime,true,sizeof(prime));
for(int i=2; i<N; i++)
{
if(prime[i])
{
p[cnt++] = i;
for(int j=i+i; j<N; j+=i)
prime[j] = false;
}
}
}

LL quick_mod(LL a,LL b
4000
,LL m)
{
LL ans = 1;
a %= m;
while(b)
{
if(b & 1)
{
ans = ans * a % m;
b--;
}
b >>= 1;
a = a * a % m;
}
return ans;
}

LL Work(LL n,LL p)
{
LL ans = 0;
while(n)
{
ans += n / p;
n /= p;
}
return ans;
}

LL Solve(LL n,LL m,LL P)
{
LL ans = 1;
for(int i=0; i<cnt && p[i]<=n; i++)
{
LL x = Work(n, p[i]);
LL y = Work(n - m, p[i]);
LL z = Work(m, p[i]);
x -= (y + z);
ans *= quick_mod(p[i],x,P);
ans %= P;
}
return ans;
}

int main()
{
int T;
isprime();
cin>>T;
while(T--)
{
LL n,m,P;
cin>>n>>m>>P;
n += m - 2;
m--;
cout<<Solve(n,m,P)<<endl;
}
return 0;
}

FFT：

//china no.1
#pragma comment(linker, "/STACK:1024000000,1024000000")
#include <vector>
#include <iostream>
#include <string>
#include <map>
#include <stack>
#include <cstring>
#include <queue>
#include <list>
#include <stdio.h>
#include <set>
#include <algorithm>
#include <cstdlib>
#include <cmath>
#include <iomanip>
#include <cctype>
#include <sstream>
#include <functional>
#include <stdlib.h>
#include <time.h>
#include <bitset>
#include <complex>
using namespace std;

#define pi acos(-1)
#define PI acos(-1)
#define endl '\n'
#define srand() srand(time(0));
#define me(x,y) memset(x,y,sizeof(x));
#define foreach(it,a) for(__typeof((a).begin()) it=(a).begin();it!=(a).end();it++)
#define close() ios::sync_with_stdio(0); cin.tie(0);
#define FOR(x,n,i) for(int i=x;i<=n;i++)
#define FOr(x,n,i) for(int i=x;i<n;i++)
#define W while
#define sgn(x) ((x) < 0 ? -1 : (x) > 0)
#define bug printf("***********\n");
#define db double
#define ll long long
typedef long long LL;
const int INF=0x3f3f3f3f;
const LL LINF=0x3f3f3f3f3f3f3f3fLL;
const int dx[]={-1,0,1,0,1,-1,-1,1};
const int dy[]={0,1,0,-1,-1,1,-1,1};
const int maxn=1e3+10;
const int maxx=1<<17;
const double EPS=1e-8;
const double eps=1e-8;
const int mod=1e9+7;
template<class T>inline T min(T a,T b,T c) { return min(min(a,b),c);}
template<class T>inline T max(T a,T b,T c) { return max(max(a,b),c);}
template<class T>inline T min(T a,T b,T c,T d) { return min(min(a,b),min(c,d));}
template<class T>inline T max(T a,T b,T c,T d) { return max(max(a,b),max(c,d));}
template <class T>
inline bool scan_d(T &ret){char c;int sgn;if (c = getchar(), c == EOF){return 0;}
while (c != '-' && (c < '0' || c > '9')){c = getchar();}sgn = (c == '-') ? -1 : 1;ret = (c == '-') ? 0 : (c - '0');
while (c = getchar(), c >= '0' && c <= '9'){ret = ret * 10 + (c - '0');}ret *= sgn;return 1;}

inline bool scan_lf(double &num){char in;double Dec=0.1;bool IsN=false,IsD=false;in=getchar();if(in==EOF) return false;
while(in!='-'&&in!='.'&&(in<'0'||in>'9'))in=getchar();if(in=='-'){IsN=true;num=0;}else if(in=='.'){IsD=true;num=0;}
else num=in-'0';if(!IsD){while(in=getchar(),in>='0'&&in<='9'){num*=10;num+=in-'0';}}
if(in!='.'){if(IsN) num=-num;return true;}else{while(in=getchar(),in>='0'&&in<='9'){num+=Dec*(in-'0');Dec*=0.1;}}
if(IsN) num=-num;return true;}

void Out(LL a){if(a < 0) { putchar('-'); a = -a; }if(a >= 10) Out(a / 10);putchar(a % 10 + '0');}
void print(LL a){ Out(a),puts("");}
//freopen( "in.txt" , "r" , stdin );
//freopen( "data.txt" , "w" , stdout );
//cerr << "run time is " << clock() << endl;

typedef complex<long double>Complex;
/*
* 进行FFT和IFFT前的反转变换。
* 位置i和 （i二进制反转后位置）互换
* len必须去2的幂
*/
void reverse(Complex y[],int len)
{
int i,j,k;
for(i = 1, j = len/2;i < len-1; i++)
{
if(i < j)swap(y[i],y[j]);
//交换互为小标反转的元素，i<j保证交换一次
//i做正常的+1，j左反转类型的+1,始终保持i和j是反转的
k = len/2;
while( j >= k)
{
j -= k;
k /= 2;
}
if(j < k) j += k;
}
}
/*
* 做FFT
* len必须为2^k形式，
* on==1时是DFT，on==-1时是IDFT
*/
void fft(Complex a[],int n,int on)
{
reverse(a,n);
for(int i=1;i<n;i<<=1)
{
Complex wn=Complex(cos(PI/i),on*sin(PI/i));
for(int j=0;j<n;j+=(i<<1))
{
Complex w=Complex(1,0);
for(int k=0;k<i;k++,w=w*wn)
{
Complex x=a[j+k],y=w*a[j+k+i];
a[j+k]=x+y;
a[j+k+i]=x-y;
}
}
}
if(on==-1)
for(int i=0;i<n;i++)
a[i].real()/=n;
}
struct node
{
int c[maxx];
}A,B,C;
Complex a[maxx],b[maxx],c[maxx];
int main()
{
int n,x;
W(scan_d(n))
{
FOr(0,n,i)
{
scan_d(x);
x+=20000;
A.c[x]++;
B.c[x+x]++;
C.c[x+x+x]++;
}
FOR(0,maxx,i)
{
a[i]=A.c[i],b[i]=B.c[i];
}
fft(a,maxx,1);
fft(b,maxx,1);
FOR(0,maxx,i)
{
c[i]=a[i]*(a[i]*a[i]-3.0l*b[i]);
}
fft(c,maxx,-1);
FOR(0,maxx,i)
{
LL ans=((LL)(c[i].real()+0.5)+2*C.c[i])/6;
if(ans>0)
printf("%d : %lld\n",i-60000,ans);
}
}
}

因子分解--3：

LL F(LL a,LL i)
{
LL l=1,r=a,b;
W(l<r)
{
LL mid=(l+r+1)/2;
b=a;
FOR(1,i,j) b/=mid;
if(b) l=mid;
else r=mid-1;
}
return l;
}
void solve()
{
FOR(1,i,j)
{
LL root=F(n,j);
}
}

一个数因子数和：

#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <map>
#include <algorithm>
using namespace std;

#define pi acos(-1)
#define endl '\n'
#define srand() srand(time(0));
#define me(x,y) memset(x,y,sizeof(x));
#define foreach(it,a) for(__typeof((a).begin()) it=(a).begin();it!=(a).end();it++)
#define close() ios::sync_with_stdio(0); cin.tie(0);
#define FOR(x,n,i) for(int i=x;i<=n;i++)
#define FOr(x,n,i) for(int i=x;i<n;i++)
#define W while
#define sgn(x) ((x) < 0 ? -1 : (x) > 0)
#define bug printf("***********\n");
typedef long long LL;
const int INF=0x3f3f3f3f;
const LL LINF=1e18+7;
const int dx[]= {-1,0,1,0,1,-1,-1,1};
const int dy[]= {0,1,0,-1,-1,1,-1,1};
const int maxn=2005;
const int maxx=1e5+100;
const double EPS=1e-7;
//const int mod=998244353;
template<class T>inline T min(T a,T b,T c)
{
return min(min(a,b),c);
}
template<class T>inline T max(T a,T b,T c)
{
return max(max(a,b),c);
}
template<class T>inline T min(T a,T b,T c,T d)
{
return min(min(a,b),min(c,d));
}
template<class T>inline T max(T a,T b,T c,T d)
{
return max(max(a,b),max(c,d));
}
inline LL Scan()
{
LL Res=0,ch,Flag=0;
if((ch=getchar())=='-')Flag=1;
else if(ch>='0' && ch<='9')Res=ch-'0';
while((ch=getchar())>='0'&&ch<='9')Res=Res*10+ch-'0';
return Flag ? -Res : Res;
}

LL a,b,mod=9901;
LL ans=1;
LL tot,pr[maxx],vis[maxx];
void init()//素数表  O(nsqrt(n))
{
for(int i=2;i<maxx;i++)
{
if(!vis[i])pr[tot++]=i;
for(int j=0;j<tot&&i*pr[j]<maxx;j++)
{
vis[i*pr[j]]=1;
if(i%pr[j]==0)break;
}
}
}
LL _pow(LL a,LL n)
{
LL ret=1;
while(n)
{
if(n&1)  ret=ret*a%mod;
a=a*a%mod;
n>>=1;
}
return ret;
}
LL inv(LL x)
{
return _pow(x,mod-2);
}
void solve()
{
for(int i=0;i<tot;i++)
{
int c=0;
if(a%pr[i]==0)
{
while(a%pr[i]==0)
{
c++; a/=pr[i];
//	cout<<c<<" "<<pr[i]<<endl;
}
if(pr[i]%mod==0) continue;
else if(pr[i]%mod==1)
{
ans=(ans*(c*b+1))%mod;
}
else
{
LL ret=(_pow(pr[i],(c*b+1))-1+mod)%mod;
ans=(ans*ret*inv(pr[i]-1))%mod;
}
}
}
if(a>1)
{
if(a%mod==0)
return ;
else if(a%mod==1)
{
ans=(ans*(b+1))%mod;
}
else
{
LL ret=(_pow(a,(b+1))-1+mod)%mod;
ans=(ans*ret*inv(a-1))%mod;
}
}
}
int main()
{
init();
while(~scanf("%lld%lld",&a,&b))
{
ans=1;
solve();
cout<<ans<<endl;
}
}

baby-step A^x=B(mod c)

#include<iostream>
#include<cstdio>
#include<cstring>
#include<algorithm>
#include<cmath>
#define LL __int64
#define N 1000000
using namespace std;
struct Node
{
int idx;
int val;
} baby
;
bool cmp(Node n1,Node n2)
{
return n1.val!=n2.val?n1.val<n2.val:n1.idx<n2.idx;
}
int gcd(int a,int b)
{
return b==0?a:gcd(b,a%b);
}
int extend_gcd(int a,int b,int &x,int &y)
{
if(b==0)
{
x=1;
y=0;
return a;
}
int gcd=extend_gcd(b,a%b,x,y);
int t=x;
x=y;
y=t-a/b*y;
return gcd;
}
int inval(int a,int b,int n)
{
int e,x,y;
extend_gcd(a,n,x,y);
e=((LL)x*b)%n;
return e<0?e+n:e;
}
int PowMod(int a,int b,int MOD)
{
LL ret=1%MOD,t=a%MOD;
while(b)
{
if(b&1)
ret=((LL)ret*t)%MOD;
t=((LL)t*t)%MOD;
b>>=1;
}
return (int)ret;
}
int BinSearch(int num,int m)
{
int low=0,high=m-1,mid;
while(low<=high)
{
mid=(low+high)>>1;
if(baby[mid].val==num)
return baby[mid].idx;
if(baby[mid].val<num)
low=mid+1;
else
high=mid-1;
}
return -1;
}
int BabyStep(int A,int B,int C)
{
LL tmp,D=1%C;
int temp;
for(int i=0,tmp=1%C; i<100; i++,tmp=((LL)tmp*A)%C)
if(tmp==B)
return i;
int d=0;
while((temp=gcd(A,C))!=1)
{
if(B%temp) return -1;
d++;
C/=temp;
B/=temp;
D=((A/temp)*D)%C;
}
int m=(int)ceil(sqrt((double)C));
for(int i=0,tmp=1%C; i<=m; i++,tmp=((LL)tmp*A)%C)
{
baby[i].idx=i;
baby[i].val=tmp;
}
sort(baby,baby+m+1,cmp);
int cnt=1;
for(int i=1; i<=m; i++)
if(baby[i].val!=baby[cnt-1].val)
baby[cnt++]=baby[i];
int am=PowMod(A,m,C);
for(int i=0; i<=m; i++,D=((LL)(D*am))%C)
{
int tmp=inval(D,B,C);
if(tmp>=0)
{
int pos=BinSearch(tmp,cnt);
if(pos!=-1)
return i*m+pos+d;
}
}
return -1;
}
int main()
{
int A,B,C;
while(scanf("%d%d%d",&A,&C,&B)!=EOF)
{
if(B>=C)
{
puts("Orz,I can’t find D!");
continue;
}
int ans=BabyStep(A,B,C);
if(ans==-1)
puts("Orz,I can’t find D!");
else
printf("%d\n",ans);
}
return 0;
}

莫比乌斯：

int tot,p[maxx],miu[maxx],v[maxx];
void Mobius(int n)
{
int i,j;
for(miu[1]=1,i=2;i<=n;i++)
{
if(!v[i]) p[tot++]=i,miu[i]=-1;
for(j=0;j<tot&&i*p[j]<=n;j++)
{
v[i*p[j]]=1;
if(i%p[j]) miu[i*p[j]]=-miu[i];
else break;
}
}
}

递推式

#include <cstdio>
#include <cstring>
#include <cmath>
#include <algorithm>
#include <vector>
#include <string>
#include <map>
#include <set>
#include <cassert>
using namespace std;
#define rep(i,a,n) for (long long i=a;i<n;i++)
#define per(i,a,n) for (long long i=n-1;i>=a;i--)
#define pb push_back
#define mp make_pair
#define all(x) (x).begin(),(x).end()
#define fi first
#define se second
#define SZ(x) ((long long)(x).size())
typedef vector<long long> VI;
typedef long long ll;
typedef pair<long long,long long> PII;
const ll mod=1e9+7;
ll powmod(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;}
// head

long long _,n;
namespace linear_seq
{
const long long N=10010;
ll res
,base
,_c
,_md
;

vector<long long> Md;
void mul(ll *a,ll *b,long long k)
{
rep(i,0,k+k) _c[i]=0;
rep(i,0,k) if (a[i]) rep(j,0,k)
_c[i+j]=(_c[i+j]+a[i]*b[j])%mod;
for (long long i=k+k-1;i>=k;i--) if (_c[i])
rep(j,0,SZ(Md)) _c[i-k+Md[j]]=(_c[i-k+Md[j]]-_c[i]*_md[Md[j]])%mod;
rep(i,0,k) a[i]=_c[i];
}
long long solve(ll n,VI a,VI b)
{ // a 系数 b 初值 b[n+1]=a[0]*b
+...
//        printf("%d\n",SZ(b));
ll ans=0,pnt=0;
long long k=SZ(a);
assert(SZ(a)==SZ(b));
rep(i,0,k) _md[k-1-i]=-a[i];_md[k]=1;
Md.clear();
rep(i,0,k) if (_md[i]!=0) Md.push_back(i);
rep(i,0,k) res[i]=base[i]=0;
res[0]=1;
while ((1ll<<pnt)<=n) pnt++;
for (long long p=pnt;p>=0;p--)
{
mul(res,res,k);
if ((n>>p)&1)
{
for (long long i=k-1;i>=0;i--) res[i+1]=res[i];res[0]=0;
rep(j,0,SZ(Md)) res[Md[j]]=(res[Md[j]]-res[k]*_md[Md[j]])%mod;
}
}
rep(i,0,k) ans=(ans+res[i]*b[i])%mod;
if (ans<0) ans+=mod;
return ans;
}
VI BM(VI s)
{
VI C(1,1),B(1,1);
long long L=0,m=1,b=1;
rep(n,0,SZ(s))
{
ll d=0;
rep(i,0,L+1) d=(d+(ll)C[i]*s[n-i])%mod;
if (d==0) ++m;
else if (2*L<=n)
{
VI T=C;
ll c=mod-d*powmod(b,mod-2)%mod;
while (SZ(C)<SZ(B)+m) C.pb(0);
rep(i,0,SZ(B)) C[i+m]=(C[i+m]+c*B[i])%mod;
L=n+1-L; B=T; b=d; m=1;
}
else
{
ll c=mod-d*powmod(b,mod-2)%mod;
while (SZ(C)<SZ(B)+m) C.pb(0);
rep(i,0,SZ(B)) C[i+m]=(C[i+m]+c*B[i])%mod;
++m;
}
}
return C;
}
long long gao(VI a,ll n)
{
VI c=BM(a);
c.erase(c.begin());
rep(i,0,SZ(c)) c[i]=(mod-c[i])%mod;
return solve(n,c,VI(a.begin(),a.begin()+SZ(c)));
}
};

int main()
{
for (scanf("%I64d",&_);_;_--)
{
scanf("%I64d",&n);
printf("%I64d\n",linear_seq::gao(VI{31,197,1255,7997},n-2));
}
}

求一个数的所有因子和

#include <cstdio>
const int maxn = 500000+2;
int sum[maxn];
int main()
{
for(int i=1; i<maxn; i++)
for(int j=1; i*j<maxn; j++)
sum[i*j] += i;
int a;
while(scanf("%d",&a)!=EOF)
printf("%d\n",sum[a]);
return 0;
}

1-n 素数个数

//china no.1
#pragma comment(linker, "/STACK:1024000000,1024000000")
#include <vector>
#include <iostream>
#include <string>
#include <map>
#include <stack>
#include <cstring>
#include <queue>
#include <list>
#include <stdio.h>
#include <set>
#include <algorithm>
#include <cstdlib>
#include <cmath>
#include <iomanip>
#include <cctype>
#include <sstream>
#include <functional>
#include <stdlib.h>
#include <time.h>
#include <bitset>
#include <complex>
using namespace std;

//#define pi acos(-1)
#define PI acos(-1)
#define endl '\n'
#define srand() srand(time(0));
#define me(x,y) memset(x,y,sizeof(x));
#define foreach(it,a) for(__typeof((a).begin()) it=(a).begin();it!=(a).end();it++)
#define close() ios::sync_with_stdio(0); cin.tie(0);
#define FOR(x,n,i) for(int i=x;i<=n;i++)
#define FOr(x,n,i) for(int i=x;i<n;i++)
#define W while
#define sgn(x) ((x) < 0 ? -1 : (x) > 0)
#define bug printf("***********\n");
#define db double
#define ll long long
typedef long long LL;
const int INF=0x3f3f3f3f;
const LL LINF=0x3f3f3f3f3f3f3f3fLL;
const int dx[]={-1,0,1,0,1,-1,-1,1};
const int dy[]={0,1,0,-1,-1,1,-1,1};
const int maxn=1e3+10;
const int maxx=1e5+10;
const double EPS=1e-8;
const double eps=1e-8;
const LL mod=1e6+3;
template<class T>inline T min(T a,T b,T c) { return min(min(a,b),c);}
template<class T>inline T max(T a,T
16edb
b,T c) { return max(max(a,b),c);}
template<class T>inline T min(T a,T b,T c,T d) { return min(min(a,b),min(c,d));}
template<class T>inline T max(T a,T b,T c,T d) { return max(max(a,b),max(c,d));}
template <class T>
inline bool scan_d(T &ret){char c;int sgn;if (c = getchar(), c == EOF){return 0;}
while (c != '-' && (c < '0' || c > '9')){c = getchar();}sgn = (c == '-') ? -1 : 1;ret = (c == '-') ? 0 : (c - '0');
while (c = getchar(), c >= '0' && c <= '9'){ret = ret * 10 + (c - '0');}ret *= sgn;return 1;}

inline bool scan_lf(double &num){char in;double Dec=0.1;bool IsN=false,IsD=false;in=getchar();if(in==EOF) return false;
while(in!='-'&&in!='.'&&(in<'0'||in>'9'))in=getchar();if(in=='-'){IsN=true;num=0;}else if(in=='.'){IsD=true;num=0;}
else num=in-'0';if(!IsD){while(in=getchar(),in>='0'&&in<='9'){num*=10;num+=in-'0';}}
if(in!='.'){if(IsN) num=-num;return true;}else{while(in=getchar(),in>='0'&&in<='9'){num+=Dec*(in-'0');Dec*=0.1;}}
if(IsN) num=-num;return true;}

void Out(LL a){if(a < 0) { putchar('-'); a = -a; }if(a >= 10) Out(a / 10);putchar(a % 10 + '0');}
void print(LL a){ Out(a),puts("");}
//freopen( "in.txt" , "r" , stdin );
//freopen( "data.txt" , "w" , stdout );
//cerr << "run time is " << clock() << endl;

const int N = 5e6 + 2;
bool np
;
int prime
, pi
;
int getprime()
{
int cnt = 0;
np[0] = np[1] = true;
pi[0] = pi[1] = 0;
for(int i = 2; i < N; i++)
{
if(!np[i]) prime[++cnt] = i;
pi[i] = cnt;
for(int j = 1; j <= cnt && i * prime[j] < N; j++)
{
np[i * prime[j]] = true;
if(i % prime[j] == 0)   break;
}
}
return cnt;
}
const int M = 7;
const int PM = 2 * 3 * 5 * 7 * 11 * 13 * 17;
int phi[PM + 1][M + 1], sz[M + 1];
void init()
{
getprime();
sz[0] = 1;
for(int i = 0; i <= PM; i++)
phi[i][0] = i;
for(int i = 1; i <= M; i++)
{
sz[i] = prime[i] * sz[i - 1];
for(int j = 1; j <= PM; j++)
phi[j][i] = phi[j][i - 1] - phi[j / prime[i]][i - 1];
}
}
int sqrt2(LL x)
{
LL r = (LL)sqrt(x - 0.1);
while(r * r <= x)   ++r;
return int(r - 1);
}
int sqrt3(LL x)
{
LL r = (LL)cbrt(x - 0.1);
while(r * r * r <= x)   ++r;
return int(r - 1);
}
LL get_phi(LL x, int s)
{
if(s == 0)  return x;
if(s <= M)  return phi[x % sz[s]][s] + (x / sz[s]) * phi[sz[s]][s];
if(x <= prime[s]*prime[s])   return pi[x] - s + 1;
if(x <= prime[s]*prime[s]*prime[s] && x < N)
{
int s2x = pi[sqrt2(x)];
LL ans = pi[x] - (s2x + s - 2) * (s2x - s + 1) / 2;
for(int i = s + 1; i <= s2x; i++) ans += pi[x / prime[i]];
return ans;
}
return get_phi(x, s - 1) - get_phi(x / prime[s], s - 1);
}
LL getpi(LL x)
{
if(x < N)   return pi[x];
LL ans = get_phi(x, pi[sqrt3(x)]) + pi[sqrt3(x)] - 1;
for(int i = pi[sqrt3(x)] + 1, ed = pi[sqrt2(x)]; i <= ed; i++)
ans -= getpi(x / prime[i]) - i + 1;
return ans;
}
LL lehmer_pi(LL x)
{
if(x < N)   return pi[x];
int a = (int)lehmer_pi(sqrt2(sqrt2(x)));
int b = (int)lehmer_pi(sqrt2(x));
int c = (int)lehmer_pi(sqrt3(x));
LL sum = get_phi(x, a) + (LL)(b + a - 2) * (b - a + 1) / 2;
for (int i = a + 1; i <= b; i++)
{
LL w = x / prime[i];
sum -= lehmer_pi(w);
if (i > c) continue;
LL lim = lehmer_pi(sqrt2(w));
for (int j = i; j <= lim; j++)
sum -= lehmer_pi(w / prime[j]) - (j - 1);
}
return sum;
}
int main()
{
init();
LL n;
while(~scanf("%I64d",&n))
{
print(lehmer_pi(n));
}
return 0;
}

大数：

#include <iostream>
#include <cstring>
#include <stdio.h>
using namespace std;
#define DIGIT   4      //四位隔开,即万进制
#define DEPTH   10000        //万进制
#define MAX     25100    //题目最大位数/4，要不大直接设为最大位数也行
typedef int bignum_t[MAX+1];
int read(bignum_t a,istream&is=cin)
{
char buf[MAX*DIGIT+1],ch ;
int i,j ;
memset((void*)a,0,sizeof(bignum_t));
if(!(is>>buf))return 0 ;
for(a[0]=strlen(buf),i=a[0]/2-1;i>=0;i--)
ch=buf[i],buf[i]=buf[a[0]-1-i],buf[a[0]-1-i]=ch ;
for(a[0]=(a[0]+DIGIT-1)/DIGIT,j=strlen(buf);j<a[0]*DIGIT;buf[j++]='0');
for(i=1;i<=a[0];i++)
for(a[i]=0,j=0;j<DIGIT;j++)
a[i]=a[i]*10+buf[i*DIGIT-1-j]-'0' ;
for(;!a[a[0]]&&a[0]>1;a[0]--);
return 1 ;
}

void write(const bignum_t a,ostream&os=cout)
{
int i,j ;
for(os<<a[i=a[0]],i--;i;i--)
for(j=DEPTH/10;j;j/=10)
os<<a[i]/j%10 ;
}

int comp(const bignum_t a,const bignum_t b)
{
int i ;
if(a[0]!=b[0])
return a[0]-b[0];
for(i=a[0];i;i--)
if(a[i]!=b[i])
return a[i]-b[i];
return 0 ;
}

int comp(const bignum_t a,const int b)
{
int c[12]=
{
1
}
;
for(c[1]=b;c[c[0]]>=DEPTH;c[c[0]+1]=c[c[0]]/DEPTH,c[c[0]]%=DEPTH,c[0]++);
return comp(a,c);
}

int comp(const bignum_t a,const int c,const int d,const bignum_t b)
{
int i,t=0,O=-DEPTH*2 ;
if(b[0]-a[0]<d&&c)
return 1 ;
for(i=b[0];i>d;i--)
{
t=t*DEPTH+a[i-d]*c-b[i];
if(t>0)return 1 ;
if(t<O)return 0 ;
}
for(i=d;i;i--)
{
t=t*DEPTH-b[i];
if(t>0)return 1 ;
if(t<O)return 0 ;
}
return t>0 ;
}
void add(bignum_t a,const bignum_t b)
{
int i ;
for(i=1;i<=b[0];i++)
if((a[i]+=b[i])>=DEPTH)
a[i]-=DEPTH,a[i+1]++;
if(b[0]>=a[0])
a[0]=b[0];
else
for(;a[i]>=DEPTH&&i<a[0];a[i]-=DEPTH,i++,a[i]++);
a[0]+=(a[a[0]+1]>0);
}
void add(bignum_t a,const int b)
{
int i=1 ;
for(a[1]+=b;a[i]>=DEPTH&&i<a[0];a[i+1]+=a[i]/DEPTH,a[i]%=DEPTH,i++);
for(;a[a[0]]>=DEPTH;a[a[0]+1]=a[a[0]]/DEPTH,a[a[0]]%=DEPTH,a[0]++);
}
void sub(bignum_t a,const bignum_t b)
{
int i ;
for(i=1;i<=b[0];i++)
if((a[i]-=b[i])<0)
a[i+1]--,a[i]+=DEPTH ;
for(;a[i]<0;a[i]+=DEPTH,i++,a[i]--);
for(;!a[a[0]]&&a[0]>1;a[0]--);
}
void sub(bignum_t a,const int b)
{
int i=1 ;
for(a[1]-=b;a[i]<0;a[i+1]+=(a[i]-DEPTH+1)/DEPTH,a[i]-=(a[i]-DEPTH+1)/DEPTH*DEPTH,i++);
for(;!a[a[0]]&&a[0]>1;a[0]--);
}

void sub(bignum_t a,const bignum_t b,const int c,const int d)
{
int i,O=b[0]+d ;
for(i=1+d;i<=O;i++)
if((a[i]-=b[i-d]*c)<0)
a[i+1]+=(a[i]-DEPTH+1)/DEPTH,a[i]-=(a[i]-DEPTH+1)/DEPTH*DEPTH ;
for(;a[i]<0;a[i+1]+=(a[i]-DEPTH+1)/DEPTH,a[i]-=(a[i]-DEPTH+1)/DEPTH*DEPTH,i++);
for(;!a[a[0]]&&a[0]>1;a[0]--);
}
void mul(bignum_t c,const bignum_t a,const bignum_t b)
{
int i,j ;
memset((void*)c,0,sizeof(bignum_t));
for(c[0]=a[0]+b[0]-1,i=1;i<=a[0];i++)
for(j=1;j<=b[0];j++)
if((c[i+j-1]+=a[i]*b[j])>=DEPTH)
c[i+j]+=c[i+j-1]/DEPTH,c[i+j-1]%=DEPTH ;
for(c[0]+=(c[c[0]+1]>0);!c[c[0]]&&c[0]>1;c[0]--);
}
void mul(bignum_t a,const int b)
{
int i ;
for(a[1]*=b,i=2;i<=a[0];i++)
{
a[i]*=b ;
if(a[i-1]>=DEPTH)
a[i]+=a[i-1]/DEPTH,a[i-1]%=DEPTH ;
}
for(;a[a[0]]>=DEPTH;a[a[0]+1]=a[a[0]]/DEPTH,a[a[0]]%=DEPTH,a[0]++);
for(;!a[a[0]]&&a[0]>1;a[0]--);
}

void mul(bignum_t b,const bignum_t a,const int c,const int d)
{
int i ;
memset((void*)b,0,sizeof(bignum_t));
for(b[0]=a[0]+d,i=d+1;i<=b[0];i++)
if((b[i]+=a[i-d]*c)>=DEPTH)
b[i+1]+=b[i]/DEPTH,b[i]%=DEPTH ;
for(;b[b[0]+1];b[0]++,b[b[0]+1]=b[b[0]]/DEPTH,b[b[0]]%=DEPTH);
for(;!b[b[0]]&&b[0]>1;b[0]--);
}
void div(bignum_t c,bignum_t a,const bignum_t b)
{
int h,l,m,i ;
memset((void*)c,0,sizeof(bignum_t));
c[0]=(b[0]<a[0]+1)?(a[0]-b[0]+2):1 ;
for(i=c[0];i;sub(a,b,c[i]=m,i-1),i--)
for(h=DEPTH-1,l=0,m=(h+l+1)>>1;h>l;m=(h+l+1)>>1)
if(comp(b,m,i-1,a))h=m-1 ;
else l=m ;
for(;!c[c[0]]&&c[0]>1;c[0]--);
c[0]=c[0]>1?c[0]:1 ;
}

void div(bignum_t a,const int b,int&c)
{
int i ;
for(c=0,i=a[0];i;c=c*DEPTH+a[i],a[i]=c/b,c%=b,i--);
for(;!a[a[0]]&&a[0]>1;a[0]--);
}
void sqrt(bignum_t b,bignum_t a)
{
int h,l,m,i ;
memset((void*)b,0,sizeof(bignum_t));
for(i=b[0]=(a[0]+1)>>1;i;sub(a,b,m,i-1),b[i]+=m,i--)
for(h=DEPTH-1,l=0,b[i]=m=(h+l+1)>>1;h>l;b[i]=m=(h+l+1)>>1)
if(comp(b,m,i-1,a))h=m-1 ;
else l=m ;
for(;!b[b[0]]&&b[0]>1;b[0]--);
for(i=1;i<=b[0];b[i++]>>=1);
}
int digit(const bignum_t a,const int b)
{
int i,ret ;
for(ret=a[(b-1)/DIGIT+1],i=(b-1)%DIGIT;i;ret/=10,i--);
return ret%10 ;
}
#define SGN(x) ((x)>0?1:((x)<0?-1:0))
#define ABS(x) ((x)>0?(x):-(x))

int read(bignum_t a,int&sgn,istream&is=cin)
{
char str[MAX*DIGIT+2],ch,*buf ;
int i,j ;
memset((void*)a,0,sizeof(bignum_t));
if(!(is>>str))return 0 ;
buf=str,sgn=1 ;
if(*buf=='-')sgn=-1,buf++;
for(a[0]=strlen(buf),i=a[0]/2-1;i>=0;i--)
ch=buf[i],buf[i]=buf[a[0]-1-i],buf[a[0]-1-i]=ch ;
for(a[0]=(a[0]+DIGIT-1)/DIGIT,j=strlen(buf);j<a[0]*DIGIT;buf[j++]='0');
for(i=1;i<=a[0];i++)
for(a[i]=0,j=0;j<DIGIT;j++)
a[i]=a[i]*10+buf[i*DIGIT-1-j]-'0' ;
for(;!a[a[0]]&&a[0]>1;a[0]--);
if(a[0]==1&&!a[1])sgn=0 ;
return 1 ;
}
struct bignum
{
bignum_t num ;
int sgn ;
public :
inline bignum()
{
memset(num,0,sizeof(bignum_t));
num[0]=1 ;
sgn=0 ;
}
inline int operator!()
{
return num[0]==1&&!num[1];
}
inline bignum&operator=(const bignum&a)
{
memcpy(num,a.num,sizeof(bignum_t));
sgn=a.sgn ;
return*this ;
}
inline bignum&operator=(const int a)
{
memset(num,0,sizeof(bignum_t));
num[0]=1 ;
sgn=SGN (a);
add(num,sgn*a);
return*this ;
}
;
inline bignum&operator+=(const bignum&a)
{
if(sgn==a.sgn)add(num,a.num);
else if
(sgn&&a.sgn)
{
int ret=comp(num,a.num);
if(ret>0)sub(num,a.num);
else if(ret<0)
{
bignum_t t ;
memcpy(t,num,sizeof(bignum_t));
memcpy(num,a.num,sizeof(bignum_t));
sub (num,t);
sgn=a.sgn ;
}
else memset(num,0,sizeof(bignum_t)),num[0]=1,sgn=0 ;
}
else if(!sgn)
memcpy(num,a.num,sizeof(bignum_t)),sgn=a.sgn ;
return*this ;
}
inline bignum&operator+=(const int a)
{
if(sgn*a>0)add(num,ABS(a));
else if(sgn&&a)
{
int  ret=comp(num,ABS(a));
if(ret>0)sub(num,ABS(a));
else if(ret<0)
{
bignum_t t ;
memcpy(t,num,sizeof(bignum_t));
memset(num,0,sizeof(bignum_t));
num[0]=1 ;
add(num,ABS (a));
sgn=-sgn ;
sub(num,t);
}
else memset(num,0,sizeof(bignum_t)),num[0]=1,sgn=0 ;
}
else if
(!sgn)sgn=SGN(a),add(num,ABS(a));
return*this ;
}
inline bignum operator+(const bignum&a)
{
bignum ret ;
memcpy(ret.num,num,sizeof (bignum_t));
ret.sgn=sgn ;
ret+=a ;
return ret ;
}
inline bignum operator+(const int a)
{
bignum ret ;
memcpy(ret.num,num,sizeof (bignum_t));
ret.sgn=sgn ;
ret+=a ;
return ret ;
}
inline bignum&operator-=(const bignum&a)
{
if(sgn*a.sgn<0)add(num,a.num);
else if
(sgn&&a.sgn)
{
int ret=comp(num,a.num);
if(ret>0)sub(num,a.num);
else if(ret<0)
{
bignum_t t ;
memcpy(t,num,sizeof(bignum_t));
memcpy(num,a.num,sizeof(bignum_t));
sub(num,t);
sgn=-sgn ;
}
else memset(num,0,sizeof(bignum_t)),num[0]=1,sgn=0 ;
}
else if(!sgn)add (num,a.num),sgn=-a.sgn ;
return*this ;
}
inline bignum&operator-=(const int a)
{
if(sgn*a<0)add(num,ABS(a));
else if(sgn&&a)
{
int  ret=comp(num,ABS(a));
if(ret>0)sub(num,ABS(a));
else if(ret<0)
{
bignum_t t ;
memcpy(t,num,sizeof(bignum_t));
memset(num,0,sizeof(bignum_t));
num[0]=1 ;
add(num,ABS(a));
sub(num,t);
sgn=-sgn ;
}
else memset(num,0,sizeof(bignum_t)),num[0]=1,sgn=0 ;
}
else if
(!sgn)sgn=-SGN(a),add(num,ABS(a));
return*this ;
}
inline bignum operator-(const bignum&a)
{
bignum ret ;
memcpy(ret.num,num,sizeof(bignum_t));
ret.sgn=sgn ;
ret-=a ;
return ret ;
}
inline bignum operator-(const int a)
{
bignum ret ;
memcpy(ret.num,num,sizeof(bignum_t));
ret.sgn=sgn ;
ret-=a ;
return ret ;
}
inline bignum&operator*=(const bignum&a)
{
bignum_t t ;
mul(t,num,a.num);
memcpy(num,t,sizeof(bignum_t));
sgn*=a.sgn ;
return*this ;
}
inline bignum&operator*=(const int a)
{
mul(num,ABS(a));
sgn*=SGN(a);
return*this ;
}
inline bignum operator*(const bignum&a)
{
bignum ret ;
mul(ret.num,num,a.num);
ret.sgn=sgn*a.sgn ;
return ret ;
}
inline bignum operator*(const int a)
{
bignum ret ;
memcpy(ret.num,num,sizeof (bignum_t));
mul(ret.num,ABS(a));
ret.sgn=sgn*SGN(a);
return ret ;
}
inline bignum&operator/=(const bignum&a)
{
bignum_t t ;
div(t,num,a.num);
memcpy (num,t,sizeof(bignum_t));
sgn=(num[0]==1&&!num[1])?0:sgn*a.sgn ;
return*this ;
}
inline bignum&operator/=(const int a)
{
int t ;
div(num,ABS(a),t);
sgn=(num[0]==1&&!num [1])?0:sgn*SGN(a);
return*this ;
}
inline bignum operator/(const bignum&a)
{
bignum ret ;
bignum_t t ;
memcpy(t,num,sizeof(bignum_t));
div(ret.num,t,a.num);
ret.sgn=(ret.num[0]==1&&!ret.num[1])?0:sgn*a.sgn ;
return ret ;
}
inline bignum operator/(const int a)
{
bignum ret ;
int t ;
memcpy(ret.num,num,sizeof(bignum_t));
div(ret.num,ABS(a),t);
ret.sgn=(ret.num[0]==1&&!ret.num[1])?0:sgn*SGN(a);
return ret ;
}
inline bignum&operator%=(const bignum&a)
{
bignum_t t ;
div(t,num,a.num);
if(num[0]==1&&!num[1])sgn=0 ;
return*this ;
}
inline int operator%=(const int a)
{
int t ;
div(num,ABS(a),t);
memset(num,0,sizeof (bignum_t));
num[0]=1 ;
add(num,t);
return t ;
}
inline bignum operator%(const bignum&a)
{
bignum ret ;
bignum_t t ;
memcpy(ret.num,num,sizeof(bignum_t));
div(t,ret.num,a.num);
ret.sgn=(ret.num[0]==1&&!ret.num [1])?0:sgn ;
return ret ;
}
inline int operator%(const int a)
{
bignum ret ;
int t ;
memcpy(ret.num,num,sizeof(bignum_t));
div(ret.num,ABS(a),t);
memset(ret.num,0,sizeof(bignum_t));
ret.num[0]=1 ;
add(ret.num,t);
return t ;
}
inline bignum&operator++()
{
*this+=1 ;
return*this ;
}
inline bignum&operator--()
{
*this-=1 ;
return*this ;
}
;
inline int operator>(const bignum&a)
{
return sgn>0?(a.sgn>0?comp(num,a.num)>0:1):(sgn<0?(a.sgn<0?comp(num,a.num)<0:0):a.sgn<0);
}
inline int operator>(const int a)
{
return sgn>0?(a>0?comp(num,a)>0:1):(sgn<0?(a<0?comp(num,-a)<0:0):a<0);
}
inline int operator>=(const bignum&a)
{
return sgn>0?(a.sgn>0?comp(num,a.num)>=0:1):(sgn<0?(a.sgn<0?comp(num,a.num)<=0:0):a.sgn<=0);
}
inline int operator>=(const int a)
{
return sgn>0?(a>0?comp(num,a)>=0:1):(sgn<0?(a<0?comp(num,-a)<=0:0):a<=0);
}
inline int operator<(const bignum&a)
{
return sgn<0?(a.sgn<0?comp(num,a.num)>0:1):(sgn>0?(a.sgn>0?comp(num,a.num)<0:0):a.sgn>0);
}
inline int operator<(const int a)
{
return sgn<0?(a<0?comp(num,-a)>0:1):(sgn>0?(a>0?comp(num,a)<0:0):a>0);
}
inline int operator<=(const bignum&a)
{
return sgn<0?(a.sgn<0?comp(num,a.num)>=0:1):(sgn>0?(a.sgn>0?comp(num,a.num)<=0:0):a.sgn>=0);
}
inline int operator<=(const int a)
{
return sgn<0?(a<0?comp(num,-a)>=0:1):
(sgn>0?(a>0?comp(num,a)<=0:0):a>=0);
}
inline int operator==(const bignum&a)
{
return(sgn==a.sgn)?!comp(num,a.num):0 ;
}
inline int operator==(const int a)
{
return(sgn*a>=0)?!comp(num,ABS(a)):0 ;
}
inline int operator!=(const bignum&a)
{
return(sgn==a.sgn)?comp(num,a.num):1 ;
}
inline int operator!=(const int a)
{
return(sgn*a>=0)?comp(num,ABS(a)):1 ;
}
inline int operator[](const int a)
{
return digit(num,a);
}
friend inline istream&operator>>(istream&is,bignum&a)
{
read(a.num,a.sgn,is);
return  is ;
}
friend inline ostream&operator<<(ostream&os,const bignum&a)
{
if(a.sgn<0)
os<<'-' ;
write(a.num,os);
return os ;
}
};
int main()
{
int t;
cin>>t;
while(t--)
{
int n;
cin>>n;
bignum jie,ans;
jie+=1;
for(int i=1;i<=n;i++)
{
jie*=i;
}
for(int i=1;i<=n;i++)
{
ans=ans+jie/i;
}
cout<<ans<<".0"<<endl;
}
}