#include #include typedef long long int ll; using namespace std; typedef pair P; using namespace atcoder; template using min_priority_queue = priority_queue, greater>; #define USE998244353 #ifdef USE998244353 const ll MOD = 998244353; using mint = modint998244353; #else const ll MOD = 1000000007; using mint = modint1000000007; #endif #pragma region //使いがち const int MAX = 2000001; long long fac[MAX], finv[MAX], inv[MAX]; void COMinit() { fac[0] = fac[1] = 1; finv[0] = finv[1] = 1; inv[1] = 1; for (int i = 2; i < MAX; i++){ fac[i] = fac[i - 1] * i % MOD; inv[i] = MOD - inv[MOD%i] * (MOD / i) % MOD; finv[i] = finv[i - 1] * inv[i] % MOD; } } long long COM(int n, int k){ if (n < k) return 0; if (n < 0 || k < 0) return 0; return fac[n] * (finv[k] * finv[n - k] % MOD) % MOD; } ll gcd(ll x, ll y) { if (y == 0) return x; else if (y > x) { return gcd (y, x); } else return gcd(x % y, y); } ll lcm(ll x, ll y) { return x / gcd(x, y) * y; } ll pow_ll(ll x, ll n) { if (n == 0) return 1; if (n % 2) { return pow_ll(x, n - 1) * x; } else { ll tmp = pow_ll(x, n / 2); return tmp * tmp; } } // floor(a^(1/k)) ll floor_root(ll a, ll k) { assert(a >= 0); assert(k >= 1); if (a == 0) return 0; if (k == 1) return a; // 大体の値 ll x = (ll)pow(a, 1.0 / k); // 増やす while ((pow_ll(x + 1, k)) <= a) { x++; } // 減らす while ((pow_ll(x, k)) > a) { x--; } return x; } ll keta(ll num, ll arity) { ll ret = 0; while (num) { num /= arity; ret++; } return ret; } // k進数で見た時のi桁目の数を返す (一番下は0桁目) ll keta_num(ll num, ll i, ll k) { return (num / pow_ll(k, i)) % k; } ll ceil(ll n, ll m) { // n > 0, m > 0 ll ret = n / m; if (n % m) ret++; return ret; } void compress(vector& v) { // [3 5 5 6 1 1 10 1] -> [1 2 2 3 0 0 4 0] vector u = v; sort(u.begin(), u.end()); u.erase(unique(u.begin(),u.end()),u.end()); map mp; for (int i = 0; i < u.size(); i++) { mp[u[i]] = i; } for (int i = 0; i < v.size(); i++) { v[i] = mp[v[i]]; } } vector > prime_factorize(ll N) { vector > res; for (ll a = 2; a * a <= N; ++a) { if (N % a != 0) continue; ll ex = 0; // 指数 // 割れる限り割り続ける while (N % a == 0) { ++ex; N /= a; } // その結果を push res.push_back({a, ex}); } // 最後に残った数について if (N != 1) res.push_back({N, 1}); return res; } #pragma endregion // xを何乗したらnの倍数になるか(divはnの素因数分解) ll calc(vector

& div, ll x) { ll ans = 0; for (auto pa : div) { ll p = pa.first; ll a = pa.second; ll cnt = 0; while (x % p == 0) { x /= p; cnt++; } if (cnt == 0) return -1; ans = max(ans, ceil(a, cnt)); } return ans; } // k!がnの倍数になる最小のkを求める ll get_num(ll n) { ll x = n; for (ll i = 1; i <= n; i++) { x /= gcd(x, i); if (x == 1) return i; } return 0; } // x^k <= nを満たす最大のkを求める ll get_pow(ll x, ll n) { if (x == 1) return 1e18; ll k = 1; ll cur = x; while (cur <= n / k) { k++; cur *= x; } return k; } int main() { ll n, a, b, c; cin >> n >> a >> b >> c; ll INF = 1e18; vector dis(n + 1, INF); dis[1] = 0; auto v = prime_factorize(n); ll bound = get_num(n); cerr << bound << endl; for (ll i = 1; i < n; i++) { // 操作1 dis[i + 1] = min(dis[i + 1], dis[i] + a); // 操作2 ll max_k = min(get_pow(INF, b), get_pow(n, i)); for (ll k = 2; k <= max_k; k++) { dis[pow_ll(i, k)] = min(dis[pow_ll(i, k)], dis[i] + pow_ll(b, k)); } // Nの倍数になる場合もOK ll k_reach = calc(v, i); if (k_reach != -1) { if (k_reach <= get_pow(INF, b)) { dis[n] = min(dis[n], dis[i] + k_reach * b); } } // 操作3 ll cur = 1; for (ll j = 1; j <= i; j++) { cur *= j; if (cur > n) break; } if (cur <= n) { dis[cur] = min(dis[cur], dis[i] + c); } else { // 怪しい dis[n] = min(dis[n], dis[i] + 2 * c); } // Nの倍数になる場合 if (i >= bound) { dis[n] = min(dis[n], dis[i] + c); } } cout << dis[n] << endl; return 0; }