#999352 (C++23) No.2829 GCD Divination

提出ソース
結果

問題	No.2829 GCD Divination
ユーザー	👑 bo9chan
提出日時	2024-08-02 22:33:19
言語	C++23 (gcc 12.3.0 + boost 1.83.0)
結果	TLE
実行時間	-
コード長	30,715 bytes
コンパイル時間	2,105 ms
コンパイル使用メモリ	188,876 KB
実行使用メモリ	333,576 KB
最終ジャッジ日時	2024-08-02 22:33:26
合計ジャッジ時間	6,319 ms
ジャッジサーバーID （参考情報）	judge1 / judge4
このコードへのチャレンジ
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テストケース

テストケース表示
入力	結果	実行時間実行使用メモリ
testcase_00	AC	2 ms 10,752 KB
testcase_01	AC	2 ms 5,376 KB
testcase_02	AC	367 ms 323,676 KB
testcase_03	AC	2 ms 5,376 KB
testcase_04	TLE	-
testcase_05	--	-
testcase_06	--	-
testcase_07	--	-
testcase_08	--	-
testcase_09	--	-
testcase_10	--	-
testcase_11	--	-
testcase_12	--	-
testcase_13	--	-
testcase_14	--	-
testcase_15	--	-
testcase_16	--	-
testcase_17	--	-
testcase_18	--	-
testcase_19	--	-
testcase_20	--	-
testcase_21	--	-
testcase_22	--	-
testcase_23	--	-
testcase_24	--	-
testcase_25	--	-
testcase_26	--	-
testcase_27	--	-
testcase_28	--	-
testcase_29	--	-
testcase_30	--	-
testcase_31	--	-
testcase_32	--	-
testcase_33	--	-
testcase_34	--	-
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ソースコード

raw source code
#ifdef __LOCAL
#include <mytemplate.hpp>  // プリコンパイル済みヘッダ ~/local/include/mytemplate.hpp.gch
#else
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <climits>
#include <cmath>
#include <deque>
#include <functional>
#include <iomanip>
#include <iostream> 
#include <iterator>
#include <list>
#include <map>
#include <memory>
#include <numeric>
#include <optional>
#include <random>
#include <queue>
#include <set>
#include <stack>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
#endif
using namespace std;
// #pragma GCC target "no-avx"  // gcc12.2のバグ回避用。boost使用時はコメントアウトの事
// #include <boost/multiprecision/cpp_int.hpp>
// using bint = boost::multiprecision::cpp_int;
using int128 = __int128_t;
using ll = long long;
using vl = vector<ll>;
using vvl = vector<vl>;
using vb = vector<bool>;
using vvb = vector<vb>;
using vd = vector<double>;
using vvd = vector<vd>;
using vc = vector<char>;
using vvc = vector<vc>;
using pll = pair<ll, ll>;
#define REP1(i, n) REP3(i, 0, n, 1)
#define REP2(i, s, n) REP3(i, s, n, 1)
#define REP3(i, s, n, d) for(ll i = (ll)(s); i < (ll)(n); i += (d))
#define REP_OVERLOAD(e1, e2, e3, e4, NAME,...) NAME
#define rep(...) REP_OVERLOAD(__VA_ARGS__, REP3, REP2, REP1)(__VA_ARGS__)
#define DEP1(i, n) DEP3(i, n, -1, 1)
#define DEP2(i, n, s) DEP3(i, n, s, 1)
#define DEP3(i, n, s, d) for(ll i = (ll)(n); (ll)(s) < i; i -= (d))
#define DEP_OVERLOAD(e1, e2, e3, e4, NAME,...) NAME
#define dep(...) DEP_OVERLOAD(__VA_ARGS__, DEP3, DEP2, DEP1)(__VA_ARGS__)
#define fore(e, a) for (auto&& e: (a))
#define len(a) (ll)(a).size()
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define pb push_back
#define eb emplace_back
#define fi first
#define se second
#define th third
/*********    C O N S T A N T    **********/
constexpr ll INF = LONG_LONG_MAX / 2 - 10000LL;  // 4,611,686,018,427,377,903  ~= 4.6e+18, 19桁
const double PI = acos(-1);
/*********       P A I R       **********/
template<class T, class U> inline pair<T, U>& operator+=(pair<T, U>& a, const pair<T, U> b);
template<class T, class U> inline pair<T, U>& operator-=(pair<T, U>& a, const pair<T, U> b);
template<class T, class U> inline pair<T, U>& operator*=(pair<T, U>& a, const pair<T, U> b);
template<class T, class U> inline pair<T, U>& operator/=(pair<T, U>& a, const pair<T, U> b);
template<class T, class U> inline pair<T, U>& operator%=(pair<T, U>& a, const pair<T, U> b);
template<class T, class U, class V> inline pair<T, U>& operator+=(pair<T, U>& a, const V b);
template<class T, class U, class V> inline pair<T, U>& operator-=(pair<T, U>& a, const V b);
template<class T, class U, class V> inline pair<T, U>& operator*=(pair<T, U>& a, const V b);
template<class T, class U, class V> inline pair<T, U>& operator/=(pair<T, U>& a, const V b);
template<class T, class U, class V> inline pair<T, U>& operator%=(pair<T, U>& a, const V b);
template<class T, class U, class V> inline pair<T, U> operator+(pair<T, U> a, const V& b);
template<class T, class U, class V> inline pair<T, U> operator-(pair<T, U> a, const V& b);
template<class T, class U, class V> inline pair<T, U> operator*(pair<T, U> a, const V& b);
template<class T, class U, class V> inline pair<T, U> operator/(pair<T, U> a, const V& b);
template<class T, class U, class V> inline pair<T, U> operator%(pair<T, U> a, const V& b);
template<class T, class U> inline pair<T, U> operator+(pair<T, U> a);
template<class T, class U> inline pair<T, U> operator-(pair<T, U> a);
template<class T, class U> istream& operator >>(istream& stream, pair<T, U>& a);
template<class T, class U> ostream& operator <<(ostream &stream, const pair<T, U>& a);
/*********       V E C T O R      **********/
template<class T> inline vector<T>& operator+=(vector<T>& a, const vector<T> b);
template<class T> inline vector<T>& operator-=(vector<T>& a, const vector<T> b);
template<class T> inline vector<T>& operator*=(vector<T>& a, const vector<T> b);
template<class T> inline vector<T>& operator/=(vector<T>& a, const vector<T> b);
template<class T> inline vector<T>& operator%=(vector<T>& a, const vector<T> b);
template<class T, class U> inline vector<T>& operator+=(vector<T>& a, const U b);
template<class T, class U> inline vector<T>& operator-=(vector<T>& a, const U b);
template<class T, class U> inline vector<T>& operator*=(vector<T>& a, const U b);
template<class T, class U> inline vector<T>& operator/=(vector<T>& a, const U b);
template<class T, class U> inline vector<T>& operator%=(vector<T>& a, const U b);
template<class T, class U> inline vector<T> operator+(vector<T> a, const U& b);
template<class T, class U> inline vector<T> operator-(vector<T> a, const U& b);
template<class T, class U> inline vector<T> operator*(vector<T> a, const U& b);
template<class T, class U> inline vector<T> operator/(vector<T> a, const U& b);
template<class T, class U> inline vector<T> operator%(vector<T> a, const U& b);
template<class T> inline vector<T> operator+(vector<T> a);
template<class T> inline vector<T> operator-(vector<T> a);
template<class T> istream& operator >>(istream& stream, vector<T>& a);
template<class T> ostream& operator <<(ostream& stream, const vector<T>& v);
template<class T> ostream& operator <<(ostream& stream, const vector<vector<T>>& vv);
template<class T> inline T Sum(const vector<T>& v) {return reduce(all(v));}  // v.size() == 0 のとき T() を返す
template<class T> inline T Max(const vector<T>& v) {assert(v.size()); return *max_element(all(v));}
template<class T> inline T Min(const vector<T>& v) {assert(v.size()); return *min_element(all(v));}
template<class T> inline ll Argmax(const vector<T>& v) {assert(v.size()); return max_element(all(v)) - v.begin();}
template<class T> inline ll Argmin(const vector<T>& v) {assert(v.size()); return min_element(all(v)) - v.begin();}
template<class T, class U> inline bool Contains(const vector<T>& v, const U& a) {return find(all(v), a) != v.end();}
template<class T> inline void Unique(vector<T>& v) {sort(all(v)); v.erase(unique(all(v)), v.end());}  // ソートされたユニーク値のvectorに書き換える, O(NlogN).
template<class T, class U> vector<T> make_vector(int n, U v) { return vector<T>(n, v); }
template <class T, class... Args> auto make_vector(int n, Args... args) {auto val = make_vector<T>(args...); return make_vector<decltype(val)>(n, move(val));}
/*********        S E T        **********/
template<class T> istream& operator >>(istream& stream, set<T>& st);
template<class T> ostream& operator <<(ostream& stream, const set<T>& st);
template<class T> inline T Max(const set<T>& st) {assert(st.size()); return *prev(st.end());}
template<class T> inline T Min(const set<T>& st) {assert(st.size()); return *st.begin();}
/*********   M U L T I S E T   **********/
template<class T> istream& operator >>(istream& stream, multiset<T>& st);
template<class T> ostream& operator <<(ostream& stream, const multiset<T>& st);
template<class T> inline T Max(const multiset<T>& st) {assert(st.size()); return *prev(st.end());}
template<class T> inline T Min(const multiset<T>& st) {assert(st.size()); return *st.begin();}
template<class T> inline bool EraseOne(multiset<T>& st, T x) {auto it=st.find(x); if (it!=st.end()) {st.erase(it); return true;} else return false;}  // 要素xを1つ削除. 消せたらtrueを返す.
/*********     S T R I N G     **********/
inline string ToUpper(const string& s) {string t; t.resize(s.size()); std::transform(all(s), t.begin(), ::toupper); return t; }
inline string ToLower(const string& s) {string t; t.resize(s.size()); std::transform(all(s), t.begin(), ::tolower); return t; }
inline char ToUpper(const char& c) {string s{c}, t; t.resize(s.size()); std::transform(all(s), t.begin(), ::toupper); return t[0]; }
inline char ToLower(const char& c) {string s{c}, t; t.resize(s.size()); std::transform(all(s), t.begin(), ::tolower); return t[0]; }
inline bool IsUpper(const char& c) {return isupper(c) > 0; }
inline bool IsLower(const char& c) {return islower(c) > 0; }
inline bool IsPalindrome(const string& S) {ll N=len(S); for (ll i=0; 2*i<N; i++) if (S[i]!=S[N-i-1]) return false; return true;}  // O(N), 空文字の返り値はTrue.
/*********       T R I O       **********/
template<class T1, class T2, class T3> struct trio {
    T1 first; T2 second; T3 third;
    // コンストラクタ・代入演算子
    trio() {first=T1(); second=T2(); third=T3();}
    trio(const T1& x) : first(x), second(x), third(x) {}
    trio(const T1& x, const T2& y, const T3& z) : first(x), second(y), third(z) {}
    trio(const trio& t) {first=t.first; second=t.second; third=t.third;}  // コピーコンストラクタ
    trio& operator =(const trio& t) {first=t.first; second=t.second; third=t.third; return *this;}
    // 比較演算子
    auto operator <=>(const trio&) const = default;
    // 単項演算子（+ -）
    trio operator +() const { return *this; }
    trio operator -() const { return (-1) * (*this); }
    // 複合代入演算子
    trio& operator +=(const trio& t) {first += t.first; second += t.second; third += t.third; return *this;}
    trio& operator -=(const trio& t) {first -= t.first; second -= t.second; third -= t.third; return *this;}
    trio& operator *=(const trio& t) {first *= t.first; second *= t.second; third *= t.third; return *this;}
    trio& operator /=(const trio& t) {first /= t.first; second /= t.second; third /= t.third; return *this;}
    trio& operator %=(const trio& t) {first %= t.first; second %= t.second; third %= t.third; return *this;}
    // 算術演算子
    friend trio operator +(const trio& lhs, const trio& rhs) {return trio(lhs) += rhs;}
    friend trio operator -(const trio& lhs, const trio& rhs) {return trio(lhs) -= rhs;}
    friend trio operator *(const trio& lhs, const trio& rhs) {return trio(lhs) *= rhs;}
    friend trio operator /(const trio& lhs, const trio& rhs) {return trio(lhs) /= rhs;}
    friend trio operator %(const trio& lhs, const trio& rhs) {return trio(lhs) %= rhs;}
    // 入出力
    template<class U1, class U2, class U3> friend istream& operator >>(istream&, trio<U1, U2, U3>&);
    template<class U1, class U2, class U3> friend ostream& operator <<(ostream&, const trio<U1, U2, U3>&);
    operator tuple<T1&, T2&, T3&>() {return tie(first, second, third);}
};
using tll = trio<ll, ll, ll>;
/*********    Q U A R T E T    **********/
template<class T1, class T2, class T3, class T4> struct quartet {
    T1 first; T2 second; T3 third; T4 fourth;
    // コンストラクタ・代入演算子
    quartet() {first=T1(); second=T2(); third=T3(); fourth=T4();}
    quartet(const T1& x) : first(x), second(x), third(x), fourth(x) {}
    quartet(const T1& x, const T2& y, const T3& z, const T4& w) : first(x), second(y), third(z), fourth(w) {}
    quartet(const quartet& t) {first=t.first; second=t.second; third=t.third; fourth=t.fourth;}  // コピーコンストラクタ
    quartet& operator =(const quartet& t) {first=t.first; second=t.second; third=t.third; fourth=t.fourth; return *this;}
    // 比較演算子
    auto operator <=>(const quartet&) const = default;
    // 単項演算子（+ -）
    quartet operator +() const { return *this; }
    quartet operator -() const { return (-1) * (*this); }
    // 複合代入演算子
    quartet& operator +=(const quartet& t) {first += t.first; second += t.second; third += t.third; fourth += t.fourth; return *this;}
    quartet& operator -=(const quartet& t) {first -= t.first; second -= t.second; third -= t.third; fourth -= t.fourth; return *this;}
    quartet& operator *=(const quartet& t) {first *= t.first; second *= t.second; third *= t.third; fourth *= t.fourth; return *this;}
    quartet& operator /=(const quartet& t) {first /= t.first; second /= t.second; third /= t.third; fourth /= t.fourth; return *this;}
    quartet& operator %=(const quartet& t) {first %= t.first; second %= t.second; third %= t.third; fourth %= t.fourth; return *this;}
    // 算術演算子
    friend quartet operator +(const quartet& lhs, const quartet& rhs) {return quartet(lhs) += rhs;}
    friend quartet operator -(const quartet& lhs, const quartet& rhs) {return quartet(lhs) -= rhs;}
    friend quartet operator *(const quartet& lhs, const quartet& rhs) {return quartet(lhs) *= rhs;}
    friend quartet operator /(const quartet& lhs, const quartet& rhs) {return quartet(lhs) /= rhs;}
    friend quartet operator %(const quartet& lhs, const quartet& rhs) {return quartet(lhs) %= rhs;}
    // 入出力
    template<class U1, class U2, class U3, class U4> friend istream& operator >>(istream&, quartet<U1, U2, U3, U4>&);
    template<class U1, class U2, class U3, class U4> friend ostream& operator <<(ostream&, const quartet<U1, U2, U3, U4>&);
    operator tuple<T1&, T2&, T3&, T4&>() {return tie(first, second, third, fourth);}
};
using qll = quartet<ll, ll, ll, ll>;
/*********     R A N D O M     **********/
struct Random {
    mt19937_64 rnd;
    Random() {
        random_device seed_gen;
        rnd.seed(seed_gen());
    }
    Random(ll seed) {
        rnd.seed(seed);
    }
    ll randint(ll a, ll b) {  // [a, b]
        uniform_int_distribution<ll> dist(a, b);
        return dist(rnd);
    }
    double randreal(double a, double b) {
        uniform_real_distribution<double> dist(a, b);
        return dist(rnd);
    }
    char randchar(char a, char b) {  // [a, b]
        uniform_int_distribution<ll> dist(a, b);
        return dist(rnd);
    }
    template<class T> void shuffle(vector<T>& vec) {
        std::shuffle(all(vec), rnd);
    }
};
/*********       T I M E R       **********/
struct Timer {
    int time_limit;
    int start_time;
    Timer(double time_limit_sec) {
        assert(time_limit_sec > 0);
        this->time_limit = time_limit_sec * CLOCKS_PER_SEC;
        this->start_time = clock();
    }
    bool is_time_up() const {
        return clock() - this->start_time >= this->time_limit;
    }
    double get_elapsed_ratio() const {
        double ratio = 1.0 * (clock() - this->start_time) / (this->time_limit - this->start_time);
        return min(1.0, ratio);
    }
};
/*********       P R I N T       **********/
template<class T> inline void print(const T& e);
template<class H, class... T> inline void print(const H& h, const T&... t);
template<class... T> inline void End(const T&... t);
/*********       D E B U G       **********/
#ifdef __LOCAL
#define debug(...) if(DEBUG) do{cout << '[' << #__VA_ARGS__ << "] ";debug_(__VA_ARGS__);}while(0)
#else
#define debug(...)
#endif
void dbg_(const long long& e);
template<class T> void dbg_(const T& e);
template<class T, class U> void dbg_(const pair<T, U>& p);
template<class T1, class T2, class T3> void dbg_(const trio<T1, T2, T3>& t);
template<class T1, class T2, class T3, class T4> void dbg_(const quartet<T1, T2, T3, T4>& t);
template<class T> void debug_(const T& e);
template<class T> void debug_(const vector<T>& v);
template<class T> void debug_(const vector<vector<T>>& vv);
template<class T, class U> void debug_(const map<T, U>& mp);
template<class T, class U> void debug_(const vector<map<T, U>>& vm);
template<class T> void debug_(const set<T>& st);
template<class T> void debug_(const multiset<T>& st);
template<class T> void debug_(const vector<set<T>>& vs);
template<class T> void debug_(const vector<multiset<T>>& vs);
template<class H, class... T> void debug_(const H& h, const T&... t);
/*********       O T H E R S       **********/
template<class T, class U> inline bool chmin(T& a, U b) {if (a > b) {a = b; return true;} return false;}  // bは値渡し!
template<class T, class U> inline bool chmax(T& a, U b) {if (a < b) {a = b; return true;} return false;}
template<class T, class U> inline auto Mod(const T& a, const U& m) {return (a % m + m) % m;}  // 負もOK
template<class T, class U> inline auto Ceil(const T& x, const U& y) {return x < 0 ? x/y : (x + y - 1) / y;}  // 負もOK
template<class T, class U> inline auto Floor(const T& x, const U& y) {return -(Ceil(-x, y)); }  // 負もOK
inline ll Isqrt(ll n) {assert(n >= 0); ll x = round(sqrt(n)); while(x * x > n) --x; return x;}
inline ll Comb(ll n, ll r) {if (r < 0 || n < r) return 0; r = min(r, n - r); ll ret = 1; rep(i, r) {ret *= n - i; ret /= i + 1;} return ret;}  // n=60, r=30までOK
template<class T> inline T Pow(T x, ll n) {assert(n >= 0); T ret = 1; while(1) {if (n % 2) ret *= x; n /= 2; if(!n) return ret; x = x * x;}}  // べき乗 mintもOK
template<class T> inline T Aseries(T a, T d, ll n) {assert(n >= 0); return a * n + n * (n - 1) / 2 * d;}  // 等差級数 mintもOK
template<class T> inline T Gseries(T a, T r, ll n) {assert(n >= 0); if (r == 1) return a * n; else return a * (1 - Pow(r, n)) / (1 - r);}  // 等比級数 mintもOK
template<class T> using min_priority_queue = priority_queue<T, vector<T>, greater<T>>;
inline bool Bit(ll b, int i) {assert(0 <= i && i < 64); return (b >> i) & 1;}
inline ll Popcount(ll b) {return __builtin_popcountll(b);}
inline ll Mask(ll n) {assert(0 <= n && n < 63); return (1LL << n) - 1LL;}  // [0, n)のbitが立った64bit整数を返す。
inline ll Mask(ll n, ll m) {assert(n >= m); return Mask(n) ^ Mask(m);}  // [m, n)のbitが立った64bit整数を返す。
inline pll GetOverlap(ll l1, ll r1, ll l2, ll r2) { return {max(l1, l2), min(r1, r2)}; }
inline bool IsOverlap(ll l1, ll r1, ll l2, ll r2) { return max(l1, l2) < min(r1, r2); }  // 引数が片方でも空区間ならfalse;
inline void PrintYesNo(bool b) {b ? print("Yes") : print("No");}
/*********       M  O  D       **********/
// #include <atcoder/modint>  // https://atcoder.github.io/ac-library/production/document_ja/
// using namespace atcoder;
// using mint = modint998244353;     // modint1000000007;
// istream& operator >>(istream& stream, mint& e) {ll n; stream >> n; e = n; return stream; }
// ostream& operator <<(ostream& stream, const mint& e) { stream << e.val(); return stream; }
// using vm = vector<mint>;
// using vvm = vector<vm>;
/***************************************/
constexpr bool DEBUG = true;

// 線形篩　参考：https://atcoder.jp/contests/abc292/editorial/5912
struct LinearSieve {
    ll N;
    vl primes;           // N以下の素数列
    vl spf;              // smallest prime factor (ex. spf[75] = 3)
    vl power;            // iが含むspf[i]の個数 (ex. power[40] = 3)
    vl prev_divisor;     // prev_divisor[i] = i / spf[i]^power[i].

    LinearSieve(ll N) : N(N), spf(N + 1), power(N + 1), prev_divisor(N + 1) {
        rep(i, 2, N + 1) {
            if (spf[i] == 0) {
                primes.pb(i);
                spf[i] = i;
                power[i] = 1;
                prev_divisor[i] = 1;
            }
            fore(p, primes) {
                if (p * i > N) break;
                spf[p * i] = p;
                if (p == spf[i]) {
                    prev_divisor[p * i] = prev_divisor[i];
                    power[p * i] = power[i] + 1;
                    break;
                }
                prev_divisor[p * i] = i;
                power[p * i] = 1;
            }
        }
    }

    // 素因数分解結果を取得, O(mlogm) (m: nの素因数の種類数 )
    map<ll, ll> get(ll n) {
        assert(1 <= n && n <= N);
        map<ll, ll> mp;
        while(n > 1) {
            mp[spf[n]] = power[n];
            n = prev_divisor[n];
        }
        return mp;
    }

    // 約数列挙（ソートなし）, O(M) (M: nの約数の個数)
    //参考: https://atcoder.jp/contests/abc170/submissions/14359259
    vl divisor(ll n) {
        // 約数の個数を求め、配列を初期化
        ll cnt = divisor_count(n);
        vl ret(cnt);
        ret[0] = 1;
        cnt = 1;  // 計算した約数の個数

        while(n > 1) {
            ll d = spf[n];
            rep(k, 1, power[n] + 1) {
                rep(j, cnt) {
                    ret[k * cnt + j] = ret[j] * d;
                }
                d *= spf[n];
            }
            cnt *= power[n] + 1;
            n = prev_divisor[n];
        }
        return ret;
    }

    // 約数の個数
    ll divisor_count(ll n) {
        assert(1 <= n && n <= N);
        ll ret = 1;
        while(n > 1) {
            ret *= power[n] + 1;
            n = prev_divisor[n];
        }
        return ret;
    }

    // オイラーのφ関数
    ll phi(ll n) {
        assert(1 <= n && n <= N);
        ll ret = n;
        while(n > 1) {
            ll p = spf[n];
            ret /= p;
            ret *= p - 1;
            n = prev_divisor[n];
        }
        return ret;
    }
};
// LinearSieve spf(100);          // O(N)
// auto mp = spf.get(12);         // (2, 2), (3, 1). 素因数分解, O(mlogm) (m: 素因数の種類数)
// vl vec = spf.divisor(12);      // {1, 2, 4, 3, 6, 12}. 約数列挙（ソートなし）, O(M) (M: nの約数の個数)
// ll n = spf.divisor_count(12);  // 6. 約数の個数, O(m)
// ll p = spf.phi(12);            // p = 4. オイラーのφ関数. 12と互いに素: {1, 5, 7, 11}

int main() {
    cin.tie(nullptr); cout << fixed << setprecision(12);

    ll N;
    cin >> N;
    auto spf = LinearSieve(N);


    auto dp = make_vector<double>(N + 1, 0);
    auto seen = make_vector<bool>(N + 1, false);
    auto f = [&](auto&& f, ll n) -> double {
        if (seen[n]) return dp[n];
        seen[n] = true;
        if (n == 1) return dp[n] = 0;

        double ret = 0;
        vl vec = spf.divisor(n);
        sort(rall(vec));
        vb B(n);
        fore(e, vec) {
            if (e == n || e == 1) continue;
            rep(i, 1, 10000000) {
                ll j = e * i;
                if (j >= n) break;
                if (B[j]) continue;
                ret += f(f, j);
                B[j] = true;
            }
        }
        ret /= n - 1;
        ret += 1.0 * n / (n - 1);
        return dp[n] = ret;
    };
    print(f(f, N));
}





/*********       P A I R       **********/
template<class T, class U> inline pair<T, U>& operator+=(pair<T, U>& a, const pair<T, U> b) {a.fi += b.fi; a.se += b.se; return a;}
template<class T, class U> inline pair<T, U>& operator-=(pair<T, U>& a, const pair<T, U> b) {a.fi -= b.fi; a.se -= b.se; return a;}
template<class T, class U> inline pair<T, U>& operator*=(pair<T, U>& a, const pair<T, U> b) {a.fi *= b.fi; a.se *= b.se; return a;}
template<class T, class U> inline pair<T, U>& operator/=(pair<T, U>& a, const pair<T, U> b) {a.fi /= b.fi; a.se /= b.se; return a;}
template<class T, class U> inline pair<T, U>& operator%=(pair<T, U>& a, const pair<T, U> b) {a.fi %= b.fi; a.se %= b.se; return a;}
template<class T, class U, class V> inline pair<T, U>& operator+=(pair<T, U>& a, const V b) {a.fi += b; a.se += b; return a;}
template<class T, class U, class V> inline pair<T, U>& operator-=(pair<T, U>& a, const V b) {a.fi -= b; a.se -= b; return a;}
template<class T, class U, class V> inline pair<T, U>& operator*=(pair<T, U>& a, const V b) {a.fi *= b; a.se *= b; return a;}
template<class T, class U, class V> inline pair<T, U>& operator/=(pair<T, U>& a, const V b) {a.fi /= b; a.se /= b; return a;}
template<class T, class U, class V> inline pair<T, U>& operator%=(pair<T, U>& a, const V b) {a.fi %= b; a.se %= b; return a;}
template<class T, class U, class V> inline pair<T, U> operator+(pair<T, U> a, const V& b) {a += b; return a;}
template<class T, class U, class V> inline pair<T, U> operator-(pair<T, U> a, const V& b) {a -= b; return a;}
template<class T, class U, class V> inline pair<T, U> operator*(pair<T, U> a, const V& b) {a *= b; return a;}
template<class T, class U, class V> inline pair<T, U> operator/(pair<T, U> a, const V& b) {a /= b; return a;}
template<class T, class U, class V> inline pair<T, U> operator%(pair<T, U> a, const V& b) {a %= b; return a;}
template<class T, class U> inline pair<T, U> operator+(pair<T, U> a) {return a;}
template<class T, class U> inline pair<T, U> operator-(pair<T, U> a) {return a * (-1);}
template<class T, class U> istream& operator >>(istream& stream, pair<T, U>& a) {stream >> a.fi >> a.se; return stream;}
template<class T, class U> ostream& operator <<(ostream &stream, const pair<T, U>& a) { stream << a.fi << " " << a.se; return stream; }
/*********       V E C T O R      **********/
template<class T> inline vector<T>& operator+=(vector<T>& a, const vector<T> b) {assert(a.size() == b.size()); rep(i, 0, a.size()) a[i] += b[i]; return a;}
template<class T> inline vector<T>& operator-=(vector<T>& a, const vector<T> b) {assert(a.size() == b.size()); rep(i, 0, a.size()) a[i] -= b[i]; return a;}
template<class T> inline vector<T>& operator*=(vector<T>& a, const vector<T> b) {assert(a.size() == b.size()); rep(i, 0, a.size()) a[i] *= b[i]; return a;}
template<class T> inline vector<T>& operator/=(vector<T>& a, const vector<T> b) {assert(a.size() == b.size()); rep(i, 0, a.size()) a[i] /= b[i]; return a;}
template<class T> inline vector<T>& operator%=(vector<T>& a, const vector<T> b) {assert(a.size() == b.size()); rep(i, 0, a.size()) a[i] %= b[i]; return a;}
template<class T, class U> inline vector<T>& operator+=(vector<T>& a, const U b) {fore(e, a) e += b; return a;}
template<class T, class U> inline vector<T>& operator-=(vector<T>& a, const U b) {fore(e, a) e -= b; return a;}
template<class T, class U> inline vector<T>& operator*=(vector<T>& a, const U b) {fore(e, a) e *= b; return a;}
template<class T, class U> inline vector<T>& operator/=(vector<T>& a, const U b) {fore(e, a) e /= b; return a;}
template<class T, class U> inline vector<T>& operator%=(vector<T>& a, const U b) {fore(e, a) e %= b; return a;}
template<class T, class U> inline vector<T> operator+(vector<T> a, const U& b) {a += b; return a;}
template<class T, class U> inline vector<T> operator-(vector<T> a, const U& b) {a -= b; return a;}
template<class T, class U> inline vector<T> operator*(vector<T> a, const U& b) {a *= b; return a;}
template<class T, class U> inline vector<T> operator/(vector<T> a, const U& b) {a /= b; return a;}
template<class T, class U> inline vector<T> operator%(vector<T> a, const U& b) {a %= b; return a;}
template<class T> inline vector<T> operator+(vector<T> a) {return a;}
template<class T> inline vector<T> operator-(vector<T> a) {return a * (-1);}
template<class T> istream& operator >>(istream& stream, vector<T>& a) {fore(e, a) stream >> e; return stream;}
template<class T> ostream& operator <<(ostream& stream, const vector<T>& v) {if(v.size()){stream << v[0]; rep(i, 1, v.size()) cout << " " << v[i];} return stream;}
template<class T> ostream& operator <<(ostream& stream, const vector<vector<T>>& vv) {if(vv.size()){stream << vv[0]; rep(i, 1, vv.size()) cout << '\n' << vv[i];} return stream;}
/*********        S E T        **********/
template<class T> istream& operator >>(istream& stream, set<T>& st) {T e; stream >> e; st.insert(e); return stream;}
template<class T> ostream& operator <<(ostream& stream, const set<T>& st) {if(st.size()){auto it=st.begin(); stream << *it++; for(; it!=st.end(); it++) cout << " " << *it;} return stream;}
/*********   M U L T I S E T   **********/
template<class T> istream& operator >>(istream& stream, multiset<T>& st) {T e; stream >> e; st.insert(e); return stream;}
template<class T> ostream& operator <<(ostream& stream, const multiset<T>& st) {if(st.size()){auto it=st.begin(); stream << *it++; for(; it!=st.end(); it++) cout << " " << *it;} return stream;}
/*********       P R I N T       **********/
template<class T> inline void print(const T& e) {cout << e << '\n';}
template<class H, class... T> inline void print(const H& h, const T&... t) {cout << h << ' '; print(t...);}
template<class... T> inline void End(const T&... t) {print(t...); exit(0);}
/*********       D E B U G       **********/
void dbg_(const long long& e) {if (e == INF) cout << "INF"; else if (e == -INF) cout << "-INF"; else cout << e;}
template<class T> void dbg_(const T& e) {cout << e;}
template<class T, class U> void dbg_(const pair<T, U>& p) {cout << '('; dbg_(p.first); cout << ' '; dbg_(p.second); cout << ')';}
template<class T1, class T2, class T3> void dbg_(const trio<T1, T2, T3>& t) {cout << '('; dbg_(t.first); cout << ' '; dbg_(t.second); cout << ' '; dbg_(t.third); cout << ')';}
template<class T1, class T2, class T3, class T4> void dbg_(const quartet<T1, T2, T3, T4>& t) {cout << '('; dbg_(t.first); cout << ' '; dbg_(t.second); cout << ' '; dbg_(t.third); cout << ' '; dbg_(t.fourth); cout << ')';}
template<class T> void debug_(const T& e) {dbg_(e); cout << '\n';}
template<class T> void debug_(const vector<T>& v) {if (v.size()){auto it=v.begin(); dbg_(*it++); for(; it!=v.end(); ++it){cout << ' '; dbg_(*it);}} cout << '\n';}
template<class T> void debug_(const vector<vector<T>>& vv) {cout << '\n'; ll cnt=0; for(auto&& v : vv){cout << cnt++ << ": "; debug_(v);}}
template<class T, class U> void debug_(const map<T, U>& mp) {if (mp.size()) {auto it = mp.begin(); dbg_(*it++); for(; it != mp.end(); ++it) {cout << ' '; dbg_(*it);}} cout << '\n';}
template<class T, class U> void debug_(const vector<map<T, U>>& vm){cout << '\n'; ll cnt=0; for(auto&& mp : vm){cout << cnt++ << ": "; debug_(mp);}}
template<class T> void debug_(const set<T>& st) {if(st.size()){auto it=st.begin(); dbg_(*it++); for(; it!=st.end(); ++it) {cout << ' '; dbg_(*it);}}cout << '\n';}
template<class T> void debug_(const multiset<T>& st) {if(st.size()) {auto it=st.begin(); dbg_(*it++); for(; it != st.end(); ++it) {cout << ' '; dbg_(*it);}} cout << '\n';}
template<class T> void debug_(const vector<set<T>>& vs) {cout << '\n'; ll cnt=0; for(auto&& st : vs){cout << cnt++ << ": "; debug_(st);}}
template<class T> void debug_(const vector<multiset<T>>& vs) {cout << '\n'; ll cnt=0;for(auto&& st : vs){cout << cnt++ << ": "; debug_(st);}}
template<class H, class... T> void debug_(const H& h, const T&... t) {dbg_(h); cout << ", "; debug_(t...);}
/*********       T R I O       **********/
template<class T1, class T2, class T3> istream& operator >>(istream &stream, trio<T1, T2, T3>& t) {return stream >> t.first >> t.second >> t.third;}
template<class T1, class T2, class T3> ostream& operator <<(ostream &stream, const trio<T1, T2, T3>& t) {return stream << t.first << " " << t.second << " " << t.third;}
/*********    Q U A R T E T    **********/
template<class T1, class T2, class T3, class T4> istream& operator >>(istream &stream, quartet<T1, T2, T3, T4>& t) {return stream >> t.first >> t.second >> t.third >> t.fourth;}
template<class T1, class T2, class T3, class T4> ostream& operator <<(ostream &stream, const quartet<T1, T2, T3, T4>& t) {return stream << t.first << " " << t.second << " " << t.third << " " << t.fourth;}
yukicoder

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