#line 1 "verify/number-theory/UNIT_mobius_function.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 2 "template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#line 2 "template/macro.hpp"
#define rep(i, a, b) for (int i = (a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b) - 1; i >= (a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
#define YN(b) cout << ((b) ? "YES" : "NO") << "\n";
#define Yn(b) cout << ((b) ? "Yes" : "No") << "\n";
#define yn(b) cout << ((b) ? "yes" : "no") << "\n";
#line 6 "template/template.hpp"
#line 2 "template/util.hpp"
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T>
using priority_queue_asc = priority_queue<T, vector<T>, greater<T>>;
template <class T, class S = T>
S SUM(const vector<T>& a) {
return accumulate(ALL(a), S(0));
}
template <class T1, class T2>
inline bool chmin(T1& a, T2 b) {
if (a > b) {
a = b;
return true;
}
return false;
}
template <class T1, class T2>
inline bool chmax(T1& a, T2 b) {
if (a < b) {
a = b;
return true;
}
return false;
}
template <class T1, class T2>
inline bool chmin_opt(optional<T1>& a, T2 b) {
if (!a || a > b) {
a = b;
return true;
}
return false;
}
template <class T1, class T2>
inline bool chmax_opt(optional<T1>& a, T2 b) {
if (!a || a < b) {
a = b;
return true;
}
return false;
}
template <class T>
int popcnt(T x) {
return __builtin_popcountll(x);
}
template <class T>
int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <class T>
int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#line 8 "template/template.hpp"
#line 2 "template/inout.hpp"
struct Fast {
Fast() {
cin.tie(nullptr);
ios_base::sync_with_stdio(false);
cout << fixed << setprecision(15);
}
} fast;
ostream& operator<<(ostream& os, __uint128_t x) {
char buf[40];
size_t k = 0;
while (x > 0) buf[k++] = (char)(x % 10 + '0'), x /= 10;
if (k == 0) buf[k++] = '0';
while (k) os << buf[--k];
return os;
}
ostream& operator<<(ostream& os, __int128_t x) {
return x < 0 ? (os << '-' << (__uint128_t)(-x)) : (os << (__uint128_t)x);
}
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a);
template <class T1, class T2>
istream& operator>>(istream& is, pair<T1, T2>& p) {
return is >> p.first >> p.second;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
return os << p.first << " " << p.second;
}
template <class T>
istream& operator>>(istream& is, vector<T>& a) {
for (auto& v : a) is >> v;
return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& a) {
for (auto it = a.begin(); it != a.end();) {
os << *it;
if (++it != a.end()) os << " ";
}
return os;
}
template <class T, size_t N>
ostream& operator<<(ostream& os, const array<T, N>& a) {
for (auto it = a.begin(); it != a.end();) {
os << *it;
if (++it != a.end()) os << " ";
}
return os;
}
template <class T>
ostream& operator<<(ostream& os, const set<T>& st) {
os << "{";
for (auto it = st.begin(); it != st.end();) {
os << *it;
if (++it != st.end()) os << ",";
}
os << "}";
return os;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const map<T1, T2>& mp) {
os << "{";
for (auto it = mp.begin(); it != mp.end();) {
os << it->first << ":" << it->second;
if (++it != mp.end()) os << ",";
}
os << "}";
return os;
}
void in() {}
template <typename T, class... U>
void in(T& t, U&... u) {
cin >> t;
in(u...);
}
template <class... T>
void in_zip(int n, T&... t) {
assert(n >= 0 && ((size(t) >= static_cast<size_t>(n)) && ...));
for (int i = 0; i < n; i++) in(t[i]...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T& t, const U&... u) {
cout << t;
if (sizeof...(u)) cout << sep;
out(u...);
}
template <class T, class U>
void out_opt(const optional<T>& opt, const U& fallback, ostream& os = cout) {
if (opt.has_value())
os << opt.value();
else
os << fallback;
os << "\n";
}
template <class T, class U>
void out_opt(const vector<optional<T>>& vec, const U& fallback, ostream& os = cout) {
for (auto it = vec.begin(); it != vec.end();) {
if ((*it).has_value())
os << (*it).value();
else
os << fallback;
if (++it != vec.end()) os << " ";
}
os << "\n";
}
namespace IO {
template <class T, class... U>
T read(U&&... u) {
T t = T(forward<U>(u)...);
in(t);
return t;
}
namespace Graph {
vector<vector<int>> unweighted(int n, int m, bool directed = false, int offset = 1) {
vector<vector<int>> g(n);
for (int i = 0; i < m; i++) {
int u, v;
cin >> u >> v;
u -= offset, v -= offset;
g[u].push_back(v);
if (!directed) g[v].push_back(u);
}
return g;
}
template <class T>
vector<vector<pair<int, T>>> weighted(int n, int m, bool directed = false, int offset = 1) {
vector<vector<pair<int, T>>> g(n);
for (int i = 0; i < m; i++) {
int u, v;
T w;
cin >> u >> v >> w;
u -= offset, v -= offset;
g[u].push_back({v, w});
if (!directed) g[v].push_back({u, w});
}
return g;
}
} // namespace Graph
namespace Tree {
vector<vector<int>> unweighted(int n, bool directed = false, int offset = 1) {
return Graph::unweighted(n, n - 1, directed, offset);
}
template <class T>
vector<vector<pair<int, T>>> weighted(int n, bool directed = false, int offset = 1) {
return Graph::weighted<T>(n, n - 1, directed, offset);
}
vector<vector<int>> rooted(int n, bool to_root = true, bool to_leaf = true, int offset = 1) {
vector<vector<int>> g(n);
for (int i = 1; i < n; i++) {
int p;
cin >> p;
p -= offset;
if (to_root) g[i].push_back(p);
if (to_leaf) g[p].push_back(i);
}
return g;
}
} // namespace Tree
} // namespace IO
#line 10 "template/template.hpp"
#line 2 "template/debug.hpp"
#ifdef LOCAL
#define debug 1
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define debug 0
#define show(...) true
#endif
template <class T>
void _show(int, T) {
cerr << '\n';
}
template <class T1, class T2, class... T3>
void _show(int i, const T1& a, const T2& b, const T3&... c) {
for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
#line 2 "number-theory/mobius-function.hpp"
#line 2 "math/util.hpp"
namespace Math {
template <class T>
T safe_mod(T a, T b) {
assert(b != 0);
if (b < 0) a = -a, b = -b;
a %= b;
return a >= 0 ? a : a + b;
}
template <class T>
T floor(T a, T b) {
assert(b != 0);
if (b < 0) a = -a, b = -b;
return a >= 0 ? a / b : (a + 1) / b - 1;
}
template <class T>
T ceil(T a, T b) {
assert(b != 0);
if (b < 0) a = -a, b = -b;
return a > 0 ? (a - 1) / b + 1 : a / b;
}
long long isqrt(long long n) {
if (n <= 0) return 0;
long long x = sqrt(n);
while ((__int128)(x + 1) * (x + 1) <= n) x++;
while ((__int128)x * x > n) x--;
return x;
}
long long floor_root(long long n, int k) {
assert(n >= 0);
if (n == 0) return 0;
assert(k >= 1);
if (k == 1) return n;
if (k > 64) return 1;
long long x = round(pow((long double)n, 1.0L / k));
auto check = [&](long long a) {
if (a <= 0) return true;
__int128_t p = 1;
for (int i = 0; i < k; ++i)
if ((p *= a) > n) return false;
return true;
};
while (check(x + 1)) x++;
while (!check(x)) x--;
return x;
}
unsigned long long floor_root_unsigned(unsigned long long n, int k) {
assert(k >= 1);
if (n <= 1 || k == 1) return n;
if (k >= 64) return 1;
int bits = (64 + k - 1) / k;
unsigned long long ok = 1, ng = min(n, 1ULL << bits);
auto check = [&](unsigned long long a) {
__uint128_t p = 1;
for (int i = 0; i < k; i++) {
p *= a;
if (p > n) return false;
}
return true;
};
while (ok + 1 < ng) {
unsigned long long mid = ok + (ng - ok) / 2;
(check(mid) ? ok : ng) = mid;
}
return ok;
}
// return g=gcd(a,b)
// a*x+b*y=g
// - b!=0 -> 0<=x<|b|/g
// - b=0 -> ax=g
template <class T>
T ext_gcd(T a, T b, T& x, T& y) {
T a0 = a, b0 = b;
bool sgn_a = a < 0, sgn_b = b < 0;
if (sgn_a) a = -a;
if (sgn_b) b = -b;
if (b == 0) {
x = sgn_a ? -1 : 1;
y = 0;
return a;
}
T x00 = 1, x01 = 0, x10 = 0, x11 = 1;
while (b != 0) {
T q = a / b, r = a - b * q;
x00 -= q * x01;
x10 -= q * x11;
swap(x00, x01);
swap(x10, x11);
a = b, b = r;
}
x = x00, y = x10;
if (sgn_a) x = -x;
if (sgn_b) y = -y;
if (b0 != 0) {
a0 /= a, b0 /= a;
if (b0 < 0) a0 = -a0, b0 = -b0;
T q = x >= 0 ? x / b0 : (x + 1) / b0 - 1;
x -= b0 * q;
y += a0 * q;
}
return a;
}
constexpr long long inv_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
long long a = m, b = x;
long long y0 = 0, y1 = 1;
while (b > 0) {
long long q = a / b;
swap(a -= q * b, b);
swap(y0 -= q * y1, y1);
}
if (y0 < 0) y0 += m / a;
return y0;
}
long long pow_mod(long long x, long long n, long long m) {
if (m == 1) return 0;
x = (x % m + m) % m;
long long y = 1;
while (n) {
if (n & 1) y = y * x % m;
x = x * x % m;
n >>= 1;
}
return y;
}
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = x % m;
if (y >= m) y += m;
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n>
constexpr bool is_prime = is_prime_constexpr(n);
}; // namespace Math
#line 2 "number-theory/prime-sieve.hpp"
namespace PrimeSieve {
using ll = long long;
vector<int> lpf(int n) {
assert(n >= 0);
vector<int> ret(n + 1);
for (size_t i = 0; i < ret.size(); i++) ret[i] = (int)i;
for (int p = 2; (ll)p * p <= n; p++) {
if (ret[p] != p) continue;
for (ll x = (ll)p * p;; x += p) {
if (ret[x] == x) ret[x] = p;
if (n - x < p) break;
}
}
return ret;
}
vector<int> table(int n) {
assert(n >= 0);
vector<bool> composite(n + 1, false);
for (int p = 2; (ll)p * p <= n; p += (p & 1) + 1) {
if (composite[p]) continue;
for (ll x = (ll)p * p;; x += p) {
composite[x] = true;
if (n - x < p) break;
}
}
vector<int> ps;
for (int p = 2; p <= n;) {
if (!composite[p]) ps.push_back(p);
int step = (p & 1) + 1;
if (n - p < step) break;
p += step;
}
return ps;
}
vector<vector<pair<ll, int>>> factorize(int n) {
assert(n >= 0);
vector<vector<pair<ll, int>>> factors(n + 1);
auto lp = lpf(n);
for (int x = 2; x <= n;) {
int y = x;
while (y > 1) {
int p = lp[y], e = 0;
while (y % p == 0) y /= p, e++;
factors[x].emplace_back(p, e);
}
if (x == n) break;
x++;
}
return factors;
}
}; // namespace PrimeSieve
/**
* @brief 素数篩
* @docs docs/number-theory/prime-sieve.md
*/
#line 5 "number-theory/mobius-function.hpp"
namespace MobiusFunction {
using ll = long long;
vector<int> table(int n) {
vector<int> mu(n + 1, 1);
mu[0] = 0;
auto lpf = PrimeSieve::lpf(n);
for (int x = 2; x <= n; x++) {
int p = lpf[x];
if (x / p % p == 0)
mu[x] = 0;
else
mu[x] = -mu[x / p];
}
return mu;
}
ll sum(ll n) {
if (n <= 0) return 0;
ll k = ceil(pow((long double)n, 2.0L / 3.0L));
__int128 n2 = (__int128)n * n;
auto enough = [&](ll x) { return (__int128)x * x * x >= n2; };
while (k > 1 && enough(k - 1)) k--;
while (!enough(k)) k++;
assert(k <= numeric_limits<int>::max());
int lim = (int)k;
vector<ll> small(lim + 1);
auto mu = table(lim);
for (int i = 1; i <= lim; i++) small[i] = small[i - 1] + mu[i];
ll len = n / k + (n % k != 0);
vector<ll> large(len + 1);
for (ll i = len; i >= 1; i--) {
ll x = n / i;
ll m = Math::isqrt(x);
ll v = 1;
for (ll j = 2; j <= m; j++) v -= i * j <= len ? large[i * j] : small[x / j];
for (ll j = 1; j <= m; j++) v -= (x / j - m) * (small[j] - small[j - 1]);
large[i] = v;
}
return large[1];
}
}; // namespace MobiusFunction
/**
* @brief Mobius Function
* @docs docs/number-theory/mobius-function.md
*/
#line 2 "util/xorshift.hpp"
namespace XORShift {
unsigned int xor32() {
static unsigned int x = 123456789u;
x ^= x << 13, x ^= x >> 17, x ^= x << 5;
return x;
}
unsigned long long xor64() {
static unsigned long long x = 123456789ull;
x ^= x << 13, x ^= x >> 7, x ^= x << 17;
return x;
}
}; // namespace XORShift
/**
* @brief XOR shift
*/
#line 6 "verify/number-theory/UNIT_mobius_function.test.cpp"
int main() {
assert(MobiusFunction::sum(-1) == 0);
assert(MobiusFunction::sum(0) == 0);
const int N = 1000000;
auto mu = MobiusFunction::table(N);
vector<long long> sum(N + 1);
for (int i = 1; i <= N; i++) sum[i] = sum[i - 1] + mu[i];
for (int n = 1; n <= 1000; n++) assert(MobiusFunction::sum(n) == sum[n]);
rep(_, 0, 100) {
int n = XORShift::xor32() % N + 1;
assert(MobiusFunction::sum(n) == sum[n]);
}
assert(MobiusFunction::sum(N) == sum[N]);
int A, B;
in(A, B);
out(A + B);
}
verify/number-theory/UNIT_mobius_function.test.cpp
math/util.hpp