#line 1 "test.cpp" //#pragma GCC target("avx,avx2") //#pragma GCC optimize("O3") //#pragma GCC optimize("unroll-loops") #include #ifdef LOCAL #include #define debug(...) debug_print::multi_print(#__VA_ARGS__, __VA_ARGS__) #else #define debug(...) (static_cast(0)) #endif using namespace std; using ll = long long; using ull = unsigned long long; using ld = long double; using pll = pair; using pii = pair; using vi = vector; using vvi = vector; using vvvi = vector; using vl = vector; using vvl = vector; using vvvl = vector; using vul = vector; using vpii = vector; using vpll = vector; using vs = vector; template using pq = priority_queue, greater>; #define overload4(_1, _2, _3, _4, name, ...) name #define overload3(a,b,c,name,...) name #define rep1(n) for (ll UNUSED_NUMBER = 0; UNUSED_NUMBER < (n); ++UNUSED_NUMBER) #define rep2(i, n) for (ll i = 0; i < (n); ++i) #define rep3(i, a, b) for (ll i = (a); i < (b); ++i) #define rep4(i, a, b, c) for (ll i = (a); i < (b); i += (c)) #define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__) #define rrep1(n) for(ll i = (n) - 1;i >= 0;i--) #define rrep2(i,n) for(ll i = (n) - 1;i >= 0;i--) #define rrep3(i,a,b) for(ll i = (b) - 1;i >= (a);i--) #define rrep4(i,a,b,c) for(ll i = (a) + ((b)-(a)-1) / (c) * (c);i >= (a);i -= c) #define rrep(...) overload4(__VA_ARGS__, rrep4, rrep3, rrep2, rrep1)(__VA_ARGS__) #define all1(i) begin(i) , end(i) #define all2(i,a) begin(i) , begin(i) + a #define all3(i,a,b) begin(i) + a , begin(i) + b #define all(...) overload3(__VA_ARGS__, all3, all2, all1)(__VA_ARGS__) #define sum(...) accumulate(all(__VA_ARGS__),0LL) template bool chmin(T &a, const T &b){ if(a > b){ a = b; return 1; } else return 0; } template bool chmax(T &a, const T &b){ if(a < b){ a = b; return 1; } else return 0; } template auto min(const T& a){return *min_element(all(a));} template auto max(const T& a){return *max_element(all(a));} template void in(Ts&... t); #define INT(...) int __VA_ARGS__; in(__VA_ARGS__) #define LL(...) ll __VA_ARGS__; in(__VA_ARGS__) #define STR(...) string __VA_ARGS__; in(__VA_ARGS__) #define CHR(...) char __VA_ARGS__; in(__VA_ARGS__) #define DBL(...) double __VA_ARGS__; in(__VA_ARGS__) #define LD(...) ld __VA_ARGS__; in(__VA_ARGS__) #define VEC(type, name, size) vector name(size); in(name) #define VV(type, name, h, w) vector> name(h, vector(w)); in(name) ll intpow(ll a, ll b){ ll ans = 1; while(b){if(b & 1) ans *= a; a *= a; b /= 2;} return ans;} ll modpow(ll a, ll b, ll p){ ll ans = 1; a %= p;if(a < 0) a += p;while(b){ if(b & 1) (ans *= a) %= p; (a *= a) %= p; b /= 2; } return ans; } void Yes() {cout << "Yes\n";return;} void No() {cout << "No\n";return;} void YES() {cout << "YES\n";return;} void NO() {cout << "NO\n";return;} namespace IO{ #define VOID(a) decltype(void(a)) struct setting{ setting(){cin.tie(nullptr); ios::sync_with_stdio(false);fixed(cout); cout.precision(12);}} setting; template struct P : P{}; template<> struct P<0>{}; template void i(T& t){ i(t, P<3>{}); } void i(vector::reference t, P<3>){ int a; i(a); t = a; } template auto i(T& t, P<2>) -> VOID(cin >> t){ cin >> t; } template auto i(T& t, P<1>) -> VOID(begin(t)){ for(auto&& x : t) i(x); } template void ituple(T& t, index_sequence){in(get(t)...);} template auto i(T& t, P<0>) -> VOID(tuple_size{}){ituple(t, make_index_sequence::value>{});} #undef VOID } #define unpack(a) (void)initializer_list{(a, 0)...} template void in(Ts&... t){ unpack(IO :: i(t)); } #undef unpack static const double PI = 3.1415926535897932; template struct REC { F f; REC(F &&f_) : f(forward(f_)) {} template auto operator()(Args &&...args) const { return f(*this, forward(args)...); }}; constexpr int mod = 998244353; //constexpr int mod = 1000000007; #line 2 "library/math/factorize.hpp" vector> prime_factorization(long long n) { vector> ret; int c = 0; while(n % 2 == 0) { c++; n >>= 1; } if(c) ret.emplace_back(2,c); for(long long i = 3; i * i <= n; i += 2) { c = 0; while(n % i == 0) { n /= i; c++; } if(c) ret.emplace_back(i,c); } if (n != 1) ret.emplace_back(n,1); return ret; } vector divisor(long long n) { vector ret; for(long long i = 1; i * i <= n; i++) { if (n % i == 0) { ret.push_back(i); if(i * i != n) {ret.push_back(n / i);} } } sort(ret.begin(),ret.end()); return ret; } #line 90 "test.cpp" int main() { INT(n,a,b,c); vl dp(n+1,1e18); dp[1] = 0; rep(i,1,n) { chmin(dp[i+1],dp[i]+a); if(i > 1) { ll now = i; ll cost = b; while(now * i <= n) { now *= i; cost *= b; if(cost >= 1e11) break; chmin(dp[now],dp[i] + cost); } if(i > 2 && i < 9) { ll x = 1; rep(j,1,i+1) x *= j; if(x <= n) chmin(dp[x],dp[i] + c); } } } auto ps = prime_factorization(n); ll ans = dp[n]; auto tmp = ps; int flg = 0; int thr = 0; rep(i,2,n+1) { if(!flg) { int now = i; int rem = 0; rep(j,tmp.size()) { while(tmp[j].second && now % tmp[j].first == 0) { now /= tmp[j].first; tmp[j].second--; } if(tmp[j].second) rem = 1; } if(rem == 0) { flg = 1; thr = i; } } if(flg) { debug(i,dp[i]); chmin(ans,dp[i] + c); } } debug(ans); rep(i,1,n+1) { int F = 1; int ma = 0; rep(j,ps.size()) { if(i % ps[j].first) F = 0; else { int cnt = 0; int now = i; while(now % ps[j].first == 0) { cnt++; now /= ps[j].first; } chmax(ma,(ps[j].second + cnt - 1) / cnt); } } if(F) { ll cost = 1; while(ma && cost <= 1e11) { ma--; cost *= b; } if(ma == 0) chmin(ans,dp[i] + cost); } ll cost = b; ll now = i; while(now < thr && cost <= 1e11) { cost *= b; now *= i; } if(now >= thr) chmin(ans,dp[i] + cost + c); } rep(i,3,n+1) { int now = i; ll C = 0; while(now <= 8 && now < thr) { C += c; ll nex = 1; rep(j,1,now+1) nex *= j; now = nex; } if(now < thr) C += 2 * c; chmin(ans,dp[i] + C); } tmp = ps; rep(i,2,thr) { int now = i; int rem = 0; rep(j,tmp.size()) { while(tmp[j].second && now % tmp[j].first == 0) { now /= tmp[j].first; tmp[j].second--; } if(tmp[j].second) rem = 1; } int ma = 0; rep(j,tmp.size()) { int x = ps[j].second - tmp[j].second; if(x == 0) ma = 1e9; else chmax(ma,(ps[j].second + x - 1) / x); } if(ma < (int)1e9) { ll cost = 1; while(ma && cost <= (ll)1e11) { cost *= b; ma--; } if(ma == 0) chmin(ans,dp[i] + c + cost); } } cout << ans << '\n'; }