https://nanti.jisuanke.com/t/41300
题意:求\(\sum_{i=1}^n\phi(i)\phi(j)2^{\phi(i)\phi(j)}\)
\(f_i=\sum_{k=1}^n[\phi(k)==i]\)
\(\sum_{i=1}^n\phi(i)\phi(j)2^{\phi(i)\phi(j)}\)
\(=\sum_{i=1}^n\sum_{j=1}^nf_if_jij2^{ij}\)
\(=2\sum_{i=1}^n\sum_{j=1}^if_if_jij2^{ij}-\sum_{i=1}^nf_ii2^{i^2}\)
\(=2\sum_{i=1}^nif_i\sum_{j=1}^ijf_j2^{i^2+j^2-(i-j)^2}-\sum_{i=1}^nf_ii2^{i^2}\)
\(=2\sum_{i=1}^nif_i{\sqrt 2}^{i^2}\sum_{j=1}^ijf_j{\sqrt 2}^{j^2}{\sqrt 2}^{-(i-j)^2}-\sum_{i=1}^nf_ii2^{i^2}\)
后一个\(\sum\)ntt,预处理f,\(\sqrt2\)的二次剩余html
//#pragma GCC optimize(2) //#pragma GCC optimize(3) //#pragma GCC optimize(4) //#pragma GCC optimize("unroll-loops") //#pragma comment(linker, "/stack:200000000") //#pragma GCC optimize("Ofast,no-stack-protector") //#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native") #include<bits/stdc++.h> //#include <bits/extc++.h> #define fi first #define se second #define db double #define mp make_pair #define pb push_back #define mt make_tuple #define pi acos(-1.0) #define ll long long #define vi vector<int> #define mod 998244353 #define ld long double //#define C 0.5772156649 #define ls l,m,rt<<1 #define rs m+1,r,rt<<1|1 #define pll pair<ll,ll> #define pil pair<int,ll> #define pli pair<ll,int> #define pii pair<int,int> #define ull unsigned long long #define bpc __builtin_popcount #define base 1000000000000000000ll #define fin freopen("1.in","r",stdin) #define fout freopen("a.txt","w",stdout) #define fio ios::sync_with_stdio(false);cin.tie(0) #define mr mt19937 rng(chrono::steady_clock::now().time_since_epoch().count()) inline ll gcd(ll a,ll b){return b?gcd(b,a%b):a;} inline void sub(ll &a,ll b){a-=b;if(a<0)a+=mod;} inline void add(ll &a,ll b){a+=b;if(a>=mod)a-=mod;} template<typename T>inline T const& MAX(T const &a,T const &b){return a>b?a:b;} template<typename T>inline T const& MIN(T const &a,T const &b){return a<b?a:b;} inline ll mul(ll a,ll b,ll c){return (a*b-(ll)((ld)a*b/c)*c+c)%c;} inline ll qp(ll a,ll b){ll ans=1;while(b){if(b&1)ans=ans*a%mod;a=a*a%mod,b>>=1;}return ans;} inline ll qp(ll a,ll b,ll c){ll ans=1;while(b){if(b&1)ans=mul(ans,a,c);a=mul(a,a,c),b>>=1;}return ans;} using namespace std; //using namespace __gnu_pbds; const ull ba=233; const db eps=1e-5; const ll INF=0x3f3f3f3f3f3f3f3f; const int N=100000+10,maxn=2000000+10,inf=0x3f3f3f3f; int prime[N],cnt,phi[N],f[N]; bool mark[N]; void init() { phi[1]=1; for(int i=2;i<N;i++) { if(!mark[i])prime[++cnt]=i,phi[i]=i-1; for(int j=1;j<=cnt&&i*prime[j]<N;j++) { mark[i*prime[j]]=1; if(i%prime[j]==0) { phi[i*prime[j]]=phi[i]*prime[j]; break; } phi[i*prime[j]]=phi[i]*phi[prime[j]]; } } } ll x[N<<3],y[N<<3]; int rev[N<<3]; void getrev(int bit) { for(int i=0;i<(1<<bit);i++) rev[i]=(rev[i>>1]>>1) | ((i&1)<<(bit-1)); } void ntt(ll *a,int n,int dft) { for(int i=0;i<n;i++) if(i<rev[i]) swap(a[i],a[rev[i]]); for(int step=1;step<n;step<<=1) { ll wn=qp(3,(mod-1)/(step*2)); if(dft==-1)wn=qp(wn,mod-2); for(int j=0;j<n;j+=step<<1) { ll wnk=1; for(int k=j;k<j+step;k++) { ll x=a[k]; ll y=wnk*a[k+step]%mod; a[k]=(x+y)%mod;a[k+step]=(x-y+mod)%mod; wnk=wnk*wn%mod; } } } if(dft==-1) { ll inv=qp(n,mod-2); for(int i=0;i<n;i++)a[i]=a[i]*inv%mod; } } int main() { ll ty=116195171ll; init(); int T;scanf("%d",&T); while(T--) { int n;scanf("%d",&n); memset(f,0,sizeof f); for(int i=1;i<=n;i++)f[phi[i]]++; int sz=0,len; while((1<<sz)<=n)sz++;sz++; len=(1<<sz); getrev(sz); x[0]=0;y[0]=1; for(int i=1;i<=n;i++) { ll te=qp(ty,1ll*i*i); x[i]=1ll*i*f[i]%mod*te%mod; y[i]=qp(te,mod-2); } for(int i=n+1;i<len;i++)x[i]=y[i]=0; ntt(x,len,1);ntt(y,len,1); for(int i=0;i<len;i++)x[i]=x[i]*y[i]%mod; ntt(x,len,-1); ll ans=0; for(int i=1;i<=n;i++) { add(ans,2ll*i*f[i]%mod*qp(ty,1ll*i*i)%mod*x[i]%mod); sub(ans,1ll*f[i]*f[i]%mod*i%mod*i%mod*qp(2,1ll*i*i)%mod); } printf("%lld\n",ans); } return 0; } /******************** ********************/