#ifndef LOCAL #pragma GCC optimize ("-Ofast") #pragma GCC optimize ("-unroll-loops") #endif #include using namespace std; using ll=long long; #define int ll #define rng(i,a,b) for(int i=int(a);i=int(a);i--) #define per(i,b) gnr(i,0,b) #define pb push_back #define eb emplace_back #define a first #define b second #define bg begin() #define ed end() #define all(x) x.bg,x.ed #define si(x) int(x.size()) #ifdef LOCAL #define dmp(x) cerr<<__LINE__<<" "<<#x<<" "< bool chmax(t&a,u b){if(a bool chmin(t&a,u b){if(b using vc=vector; template using vvc=vc>; using pi=pair; using vi=vc; template ostream& operator<<(ostream& os,const pair& p){ return os<<"{"< ostream& operator<<(ostream& os,const vc& v){ os<<"{"; for(auto e:v)os< void dmpr(ostream&os,const T&t,const Args&... args){ os< ostream& operator<<(ostream&os,const array&a){ return os<(all(a)); } template void print_tuple(ostream&,const T&){ } template void print_tuple(ostream&os,const T&t){ if(i)os<<","; os<(t); print_tuple(os,t); } template ostream& operator<<(ostream&os,const tuple&t){ os<<"{"; print_tuple<0,tuple,Args...>(os,t); return os<<"}"; } template void print(t x,int suc=1){ cout<>i; return i; } vi readvi(int n,int off=0){ vi v(n); rep(i,n)v[i]=read()+off; return v; } pi readpi(int off=0){ int a,b;cin>>a>>b; return pi(a+off,b+off); } template void print(const pair&p,int suc=1){ print(p.a,2); print(p.b,suc); } template void print(const vector&v,int suc=1){ rep(i,v.size()) print(v[i],i==int(v.size())-1?suc:2); } string readString(){ string s; cin>>s; return s; } template T sq(const T& t){ return t*t; } //#define CAPITAL void yes(bool ex=true){ #ifdef CAPITAL cout<<"YES"<<"\n"; #else cout<<"Yes"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } void no(bool ex=true){ #ifdef CAPITAL cout<<"NO"<<"\n"; #else cout<<"No"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } void possible(bool ex=true){ #ifdef CAPITAL cout<<"POSSIBLE"<<"\n"; #else cout<<"Possible"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } void impossible(bool ex=true){ #ifdef CAPITAL cout<<"IMPOSSIBLE"<<"\n"; #else cout<<"Impossible"<<"\n"; #endif if(ex)exit(0); #ifdef LOCAL cout.flush(); #endif } constexpr ll ten(int n){ return n==0?1:ten(n-1)*10; } const ll infLL=LLONG_MAX/3; #ifdef int const int inf=infLL; #else const int inf=INT_MAX/2-100; #endif int topbit(signed t){ return t==0?-1:31-__builtin_clz(t); } int topbit(ll t){ return t==0?-1:63-__builtin_clzll(t); } int botbit(signed a){ return a==0?32:__builtin_ctz(a); } int botbit(ll a){ return a==0?64:__builtin_ctzll(a); } int popcount(signed t){ return __builtin_popcount(t); } int popcount(ll t){ return __builtin_popcountll(t); } bool ispow2(int i){ return i&&(i&-i)==i; } ll mask(int i){ return (ll(1)< void mkuni(vc&v){ sort(all(v)); v.erase(unique(all(v)),v.ed); } ll rand_int(ll l, ll r) { //[l, r] #ifdef LOCAL static mt19937_64 gen; #else static mt19937_64 gen(chrono::steady_clock::now().time_since_epoch().count()); #endif return uniform_int_distribution(l, r)(gen); } template void myshuffle(vc&a){ rep(i,si(a))swap(a[i],a[rand_int(0,i)]); } template int lwb(const vc&v,const t&a){ return lower_bound(all(v),a)-v.bg; } vvc readGraph(int n,int m){ vvc g(n); rep(i,m){ int a,b; cin>>a>>b; //sc.read(a,b); a--;b--; g[a].pb(b); g[b].pb(a); } return g; } vvc readTree(int n){ return readGraph(n,n-1); } int gcd(int a,int b){return b?gcd(b,a%b):a;} template t pow_mod(t x,t n,t m){ t r=1; while(n){ if(n&1)r=(r*x)%m; x=(x*x)%m; n>>=1; } return r; } bool is_prime(ll n) { if (n <= 1) return false; if (n == 2) return true; if (n % 2 == 0) return false; ll d = n - 1; while (d % 2 == 0) d /= 2; for (ll a : {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}) { if (n <= a) break; ll t = d; ll y = pow_mod<__int128_t>(a, t, n); // over while (t != n - 1 && y != 1 && y != n - 1) { y = __int128_t(y) * y % n; // flow t <<= 1; } if (y != n - 1 && t % 2 == 0) { return false; } } return true; } ll pollard_single(ll n) { auto f = [&](ll x) { return (__int128_t(x) * x + 1) % n; }; if (is_prime(n)) return n; if (n % 2 == 0) return 2; ll st = 0; while (true) { st++; ll x = st, y = f(x); while (true) { ll p = gcd((y - x + n), n); if (p == 0 || p == n) break; if (p != 1) return p; x = f(x); y = f(f(y)); } } } vc pollard(ll n) { if (n == 1) return {}; ll x = pollard_single(n); if (x == n) return {x}; vc le = pollard(x); vc ri = pollard(n / x); le.insert(le.end(), ri.begin(), ri.end()); return le; } ll primitive_root(ll p) { auto v = pollard(p - 1); //while (true) { // ll g = rand_int(1, p - 1); //[1, p-1] rng(g,4,p){ bool ok = true; for (auto d : v) { ll f = (p - 1) / d; if (pow_mod<__int128_t>(g, f, p) == 1) { ok = false; break; } } if (ok) return g; } assert(0); } vc to_freq(vi a){ sort(all(a)); vc res; for(auto v:a){ if(res.empty()||res.back().a!=v)res.eb(v,0); res.back().b++; } return res; } vi enum_divs(int n){ auto f=to_freq(pollard(n)); dmp(f); vi res; auto rec=[&](auto self,int i,int d)->void{ if(i==si(f))res.pb(d); else{ rep(_,f[i].b+1){ self(self,i+1,d); d*=f[i].a; } } }; rec(rec,0,1); return res; } const int vmax=ten(6)+10; bool pri[vmax]; vi ps; int sf[vmax]; //smallest factor, sf[1] is undefined(0) void linear_sieve(){ rng(i,2,vmax) pri[i]=1; rng(i,2,vmax){ if(pri[i]){ ps.pb(i); sf[i]=i; } for(int j=0;i*ps[j] pow_cap(ull a,int n){ if(a==0)return make_pair(0,0); bool over=false; auto mult=[&](ull x,ull y){ if(x<=mask64/y)return x*y; else{ over=true; return mask64; } }; ull res=1; while(n){ if(n&1)res=mult(res,a); n>>=1; if(n)a=mult(a,a); } return make_pair(res,over); } //yosupo kth root integer //0<=a<2^64 ull kth_root_integer(ull a,int k){ if(a==0)return 0; ull z=min((ull)powl(a,(long double)1/k),mask64); while(1){ auto w=pow_cap(z,k); if(w.a>a||w.b)z--; else break; } while(z struct counting_primes{ const int n,n2,n3,n6; int s; vi vs,sum,bit; int pre; void trans(int i,int p){ int j=getidx(vs[i]/p); sum[i]-=sum[j]-pre; } void trans2(int i,int p){ int j=getidx(vs[i]/p); int z=sum[j]; if(j>=n3){ j-=n3; for(;j=0;x-=(x+1)&(-x-1)) bit[x]--; } for(int j=i;cur*ps[j]<=vs[n3];j++) upd(j,cur*ps[j],false); } counting_primes(int nn):n(nn),n2(isqrt(n)),n3(icbrt(n)),n6(kth_root_integer(n,6)){ { int v=n; while(v){ vs.pb(v); v=n/(n/v+1); } } s=vs.size(); sum.resize(s); rep(i,s)sum[i]=vs[i]-1; int idx=0; pre=0; { while(ps[idx]<=n6){ rep(i,s){ if(sq(ps[idx])>vs[i])break; trans(i,ps[idx]); } pre++; idx++; } } { bit.resize(s-n3); while(ps[idx]<=n3){ rep(i,n3){ if(sq(ps[idx])>vs[i])break; trans2(i,ps[idx]); } assert(ps[idx]<=vs[n3]); upd(idx,ps[idx],true); pre++; idx++; } per(i,s-n3){ int j=i+((i+1)&(-i-1)); if(jvs[i])break; trans(i,ps[idx]); } pre++; idx++; } } } vc buf; D ans; D multiplicative_sum(){ buf.resize(s); rep(i,s){ buf[i]=sum[i]; if(is_prime(vs[i]+1))buf[i]++; } ans=1+buf[0]; rep(i,si(ps))if((ps[i])*(ps[i]-1)<=n)dfs(i,n/(ps[i]-1));else break; return ans; } void dfs(int i,int lim){ //assert(ps[i]<=lim); ans++; if(lim>=sq(ps[i]))dfs(i,lim/ps[i]); ans+=buf[getidx(lim)]-buf[getidx(ps[i]-1)]; for(int j=i+1;(ps[j])*(ps[j]-1)<=lim;j++) dfs(j,lim/(ps[j]-1)); } int getidx(int a){ return a<=n2?s-a:n/a-1; } }; void slv(){ int n;cin>>n; counting_primes cp(n); print(cp.multiplicative_sum()); } signed main(){ cin.tie(0); ios::sync_with_stdio(0); cout<>t;rep(_,t) slv(); }