[BZOJ4913][SDOI2017]遗忘的集合

题解：

首先先弄出$f(x)$的生成函数
$$f(x)=\prod_{i=1}^{n} {{(\frac{1}{1-x^i})}}^{a[i]}$$
因为$f(x)$已知，我们考虑利用这个式子取推出$a[i]$
右边的乘法显然处理起来不方便，按照套路两边取对数
$$ln(f(x))=\sum_{i=1}^{n} {-a[i]*ln(1-x^i)}$$
然后看见$ln(1+x)$肯定是泰勒展开了，得到
$$ln(f(x))=\sum_{i=1}^{n} {-a[i]*\sum_{k=1}^{INF} {\frac{{(-1)}^{k-1}*{{{(-x^i)}^k}}}{k}}}$$
化简一下就是
$$ln(f(x))=\sum_{i=1}^{n} {a[i]*\sum_{k=1}^{INF} {\frac{{x^{ik}}}{k}}}$$
然后因为我们知道的$f(x)$的每一项,所以变成枚举x系数
令$T=ik$
$$ln(f(x))=\sum_{T=1}^{INF} {x^T \sum_{i|T}^{T}{\frac{i*a[i]}{T}}}$$
然后我们只需要从小到大枚举T，然后依旧算出每个$a[i]$并处理对之后的影响

常数太大只有50。。。不知道怎么卡

#include <bits/stdc++.h>
using namespace std;
#define rint register int
#define IL inline
#define rep(i,h,t) for (int i=h;i<=t;i++)
#define dep(i,t,h) for (int i=t;i>=h;i--)
#define me(x) memset(x,0,sizeof(x))
#define ll long long
namespace IO{
    char ss[<<],*A=ss,*B=ss;
    IL char gc()
    {
        return A==B&&(B=(A=ss)+fread(ss,,<<,stdin),A==B)?EOF:*A++;
    }
    template<class T>void read(T &x)
    {
        rint f=,c; while (c=gc(),c<||c>) if (c=='-') f=-; x=(c^);
        while (c=gc(),c>&&c<) x=(x<<)+(x<<)+(c^); x*=f;
    }
    char sr[<<],z[]; int C1=-,Z;
    template<class T>void wer(T x)
    {
        if (x<) sr[++C1]='-',x=-x;
        while (z[++Z]=x%+,x/=);
        while (sr[++C1]=z[Z],--Z);
    }
    IL void wer1()
    {
        sr[++C1]=' ';
    }
    IL void wer2()
    {
        sr[++C1]='\n';
    }
    template<class T>IL void maxa(T &x,T y) { if (x<y) x=y;}
    template<class T>IL void mina(T &x,T y) { if (x>y) x=y;}
    template<class T>IL void MAX(T x,T y){ return x>y?x:y;}
    template<class T>IL void MIN(T x,T y){ return x<y?x:y;}
};
using namespace IO;
const int N=3e5*;
const double ee=1.0000000000;
const double pi=acos(-1.0);
struct cp{
    double a,b;
    cp operator +(const cp o) const
    {
        return (cp){o.a+a,o.b+b};
    }
    cp operator -(const cp o) const
    {
        return (cp){a-o.a,b-o.b};
    }
    cp operator *(const cp o) const
    {
        return (cp){a*o.a-b*o.b,b*o.a+a*o.b};
    }
}a[N],b[N],w[N];
int n,m,l,r[N],mo,mo2,mo3,x1[N],x2[N];
int fsp(int x,int y)
{
    ll now=;
    while (y)
    {
        if (y&) now=now*x%mo;
        y>>=; x=1ll*x*x%mo;
    }
    return now;
}
void fft_init()
{
    l=; n=; while (n<=m) n<<=,l++;
    for (int i=;i<n;i++)
    {
      w[i]=(cp){cos(pi*i/n),sin(pi*i/n)};
      r[i]=(r[i/]/)|((i&)<<(l-));
    }
}
void fft_clear()
{
    rep(i,,n-) a[i].a=a[i].b=b[i].a=b[i].b=;
}
void fft(cp *a,int o)
{
    for (int i=;i<n;i++) if (i>r[i]) swap(a[i],a[r[i]]);
    for (int i=;i<n;i*=)
      for (int j=;j<n;j+=(i*))
      {
          cp *x1=a+j,*x2=a+i+j;
        for (int k=;k<i;k++,x1++,x2++)
        {
            cp W=w[n/i*k]; W.b*=o;
            cp x=*x1,y=(*x2)*W;
            *x1=x+y,*x2=x-y;
        }
      }
    if (o==-) rep(i,,n-) a[i].a/=n;
}
cp a1[N],a2[N],a3[N],b1[N],b2[N];
void getcj(int *A,int *B,int len)
{
    m=*len; fft_init();
    rep(i,,len-)
    {
      a1[i].a=A[i]/mo2,a2[i].a=A[i]%mo2;
      b1[i].a=B[i]/mo2,b2[i].a=B[i]%mo2;
    }
    fft(a1,); fft(a2,); fft(b1,); fft(b2,);
    rep(i,,n-)
    {
      a3[i]=a1[i]*b1[i];
      a1[i]=a1[i]*b2[i]+a2[i]*b1[i];
      b2[i]=a2[i]*b2[i];
    }
    fft(a3,-); fft(a1,-); fft(b2,-);
    rep(i,,len-)
    {
      B[i]=((ll)(a3[i].a+0.5)%mo*mo3+(ll)(a1[i].a+0.5)%mo*mo2+(ll)(b2[i].a+0.5))%mo;
      B[i]=(B[i]+mo)%mo;
    }
    rep(i,,n)
    {
      a1[i].a=a1[i].b=a2[i].a=a2[i].b=a3[i].a=a3[i].b=;
      b1[i].a=b1[i].b=b2[i].a=b2[i].b=;
    }
}
int C[N];
IL void getinv(int *A,int *B,int len)
{
    if (len==) {B[]=fsp(A[],mo-); return;}
    getinv(A,B,(len+)>>);
    rep(i,,len-) C[i]=A[i];
    getcj(B,C,len); getcj(B,C,len);
    rep(i,,len-) B[i]=((2ll*B[i]-C[i])%mo+mo)%mo;
}
IL void getdao(int *A,int *B,int len)
{
    for (int i=;i<len-;i++)
      B[i]=1ll*A[i+]*(i+)%mo;
    B[len-]=;
}
IL void getjf(int *A,int *B,int len)
{
    for (int i=;i<len;i++)
      B[i]=1ll*A[i-]*fsp(i,mo-)%mo;
    B[]=;
}
int C[N],D[N];
IL void getln(int *A,int *B,int len)
{
    getdao(A,C,len);
    getinv(A,D,len);
    getcj(C,D,len);
    getjf(D,B,len);
}
void getexp(int *A,int *B,int len)
{
    if (len==) {B[]=; return;}
    getexp(A,B,(len+)>>);
    getln(B,C,len);
    rep(i,,len-) C[i]=((-C[i]+A[i])%mo+mo)%mo;
    C[]=(C[]+)%mo;
    getcj(C,B,len);
}
int main()
{
    freopen("1.in","r",stdin);
    freopen("1.out","w",stdout);
    int n1;
    read(n1); read(mo); mo2=3e4; mo3=mo2*mo2;
    x1[]=;
    rep(i,,n1) read(x1[i]);
    getln(x1,x2,n1+);
    rep(i,,n1)
    {
        for (int j=;j*i<=n1;j++)
          x2[i*j]=(x2[i*j]-1ll*x2[i]*fsp(j,mo-))%mo;
    }
    int cnt=;
    rep(i,,n1) if (x2[i]) cnt++;
    wer(cnt); wer2();
    rep(i,,n1) if (x2[i]) wer(i),wer1();
    wer2();
    fwrite(sr,,C1+,stdout);
    return ;
}

#updata 12.14

使用我优秀的常数的新模板他就过了

#include <bits/stdc++.h>
using namespace std;
#define rint register int
#define IL inline
#define rep(i,h,t) for (int i=h;i<=t;i++)
#define dep(i,t,h) for (int i=t;i>=h;i--)
#define me(x) memset(x,0,sizeof(x))
#define ll long long
namespace IO{
    char ss[<<],*A=ss,*B=ss;
    IL char gc()
    {
        return A==B&&(B=(A=ss)+fread(ss,,<<,stdin),A==B)?EOF:*A++;
    }
    template<class T>void read(T &x)
    {
        rint f=,c; while (c=gc(),c<||c>) if (c=='-') f=-; x=(c^);
        while (c=gc(),c>&&c<) x=(x<<)+(x<<)+(c^); x*=f;
    }
    char sr[<<],z[]; int C1=-,Z;
    template<class T>void wer(T x)
    {
        if (x<) sr[++C1]='-',x=-x;
        while (z[++Z]=x%+,x/=);
        while (sr[++C1]=z[Z],--Z);
    }
    IL void wer1()
    {
        sr[++C1]=' ';
    }
    IL void wer2()
    {
        sr[++C1]='\n';
    }
    template<class T>IL void maxa(T &x,T y) { if (x<y) x=y;}
    template<class T>IL void mina(T &x,T y) { if (x>y) x=y;}
    template<class T>IL void MAX(T x,T y){ return x>y?x:y;}
    template<class T>IL void MIN(T x,T y){ return x<y?x:y;}
};
using namespace IO;
const int N=3e5*;
const double ee=1.0000000000;
const double pi=acos(-1.0);
struct cp{
    double a,b;
    cp operator +(const cp o) const
    {
        return (cp){o.a+a,o.b+b};
    }
    cp operator -(const cp o) const
    {
        return (cp){a-o.a,b-o.b};
    }
    cp operator *(const cp o) const
    {
        return (cp){a*o.a-b*o.b,b*o.a+a*o.b};
    }
}a[N],b[N],c[N],d[N];
int n,m,l,r[N],mo,x1[N],x2[N];
IL int fsp(int x,int y)
{
    ll now=;
    while (y)
    {
        if (y&) now=now*x%mo;
        x=1ll*x*x%mo;
        y>>=;
    }
    return now;
}
IL void clear()
{
    for (int i=;i<=n;i++) a[i].a=a[i].b=b[i].a=b[i].b=c[i].a=c[i].b=d[i].a=d[i].b=;
}
cp *w[N],tmp[N*];
int p;
IL void init()
{
    cp *now=tmp;
    for (int i=;i<=p;i<<=)
    {
        w[i]=now;
        for (int j=;j<i;j++) w[i][j]=(cp){cos(pi*j/i),sin(pi*j/i)};
        now+=i;
    }
}
IL void fft_init()
{
    l=; for (n=;n<=m;n<<=) l++;
    for (int i=;i<n;i++) r[i]=(r[i/]/)|((i&)<<(l-));
}
void fft(cp *a,int o)
{
    for (int i=;i<n;i++) if (i>r[i]) swap(a[i],a[r[i]]);
    for (int i=;i<n;i<<=)
        for (int j=;j<n;j+=(i*))
        {
          cp *x1=a+j,*x2=a+i+j,*W=w[i];
          for (int k=;k<i;k++,x1++,x2++,W++)
          {
              cp x=*x1,y=(cp){(*W).a,(*W).b*o}*(*x2);
              *x1=x+y,*x2=x-y;
          }
        }
    if (o==-) for(int i=;i<n;i++) a[i].a/=n;
}
IL void getcj(int *A,int *B,int len)
{
    rep(i,,len)
    {
        A[i]=(A[i]+mo)%mo,B[i]=(B[i]+mo)%mo;
    }
    for (int i=;i<len;i++)
    {
       a[i]=(cp){A[i]&,A[i]>>};
       b[i]=(cp){B[i]&,B[i]>>};
    }
    m=len*; fft_init();
    fft(a,); fft(b,);
    for (int i=;i<n;i++)
    {
        int j=(n-)&(n-i);
        c[j]=(cp){0.5*(a[i].a+a[j].a),0.5*(a[i].b-a[j].b)}*b[i];
        d[j]=(cp){0.5*(a[i].b+a[j].b),0.5*(a[j].a-a[i].a)}*b[i];
    }
    fft(c,); fft(d,);
    double inv=ee/n;
    rep(i,,n) c[i].a*=inv,c[i].b*=inv;
    rep(i,,n) d[i].a*=inv,d[i].b*=inv;
    rep(i,,len)
    {
        ll a1=c[i].a+0.5,a2=c[i].b+0.5;
        ll a3=d[i].a+0.5,a4=d[i].b+0.5;
        B[i]=(a1+((a2+a3)%mo<<)+((a4%mo)<<))%mo;
    }
    clear();
}
int C[N],D[N];
IL void getinv(int *A,int *B,int len)
{
    if (len==) {B[]=fsp(A[],mo-); return;}
    getinv(A,B,(len+)>>);
    rep(i,,len-) C[i]=A[i];
    getcj(B,C,len); getcj(B,C,len);
    rep(i,,len-) B[i]=((2ll*B[i]-C[i])%mo+mo)%mo;
}
IL void getdao(int *A,int *B,int len)
{
    for (int i=;i<len-;i++)
      B[i]=1ll*A[i+]*(i+)%mo;
    B[len-]=;
}
IL void getjf(int *A,int *B,int len)
{
    for (int i=;i<len;i++)
      B[i]=1ll*A[i-]*fsp(i,mo-)%mo;
    B[]=;
}
IL void getln(int *A,int *B,int len)
{
    getdao(A,C,len);
    getinv(A,D,len);
    getcj(C,D,len);
    getjf(D,B,len);
}
void getexp(int *A,int *B,int len)
{
    if (len==) {B[]=; return;}
    getexp(A,B,(len+)>>);
    getln(B,C,len);
    rep(i,,len-) C[i]=((-C[i]+A[i])%mo+mo)%mo;
    C[]=(C[]+)%mo;
    getcj(C,B,len);
}
int main()
{
    freopen("10.in","r",stdin);
    freopen("1.out","w",stdout);
    int n1;
    read(n1); read(mo);
    x1[]=;
    rep(i,,n1) read(x1[i]);
    p=(n1+)<<; init();
    getln(x1,x2,n1+);
    rep(i,,n1)
    {
        for (int j=;j*i<=n1;j++)
          x2[i*j]=(x2[i*j]-1ll*x2[i]*fsp(j,mo-))%mo;
    }
    int cnt=;
    rep(i,,n1) if (x2[i]) cnt++;
    wer(cnt); wer2();
    rep(i,,n1) if (x2[i]) wer(i),wer1();
    wer2();
    fwrite(sr,,C1+,stdout);
    return ;
}