#line 1 "verify/number-theory/LC_sum_of_multiplicative_function.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_multiplicative_function"
#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 "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 3 "modint/modint.hpp"
template <unsigned int m = 998244353>
struct ModInt {
using mint = ModInt;
static constexpr unsigned int get_mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
ModInt() : _v(0) {}
ModInt(int64_t v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs) { return *this *= rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if (is_prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto inv = Math::inv_mod(_v, umod());
return raw(inv);
}
}
friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; }
friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; }
friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; }
friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; }
friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; }
friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; }
friend istream& operator>>(istream& is, mint& x) {
int64_t v;
is >> v;
x = mint(v);
return is;
}
friend ostream& operator<<(ostream& os, const mint& x) { return os << x.val(); }
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool is_prime = Math::is_prime<m>;
};
using ModInt998244353 = ModInt<998244353>;
using ModInt1000000007 = ModInt<1000000007>;
#line 5 "verify/number-theory/LC_sum_of_multiplicative_function.test.cpp"
using mint = ModInt<469762049>;
#line 2 "number-theory/lucy-dp.hpp"
#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/lucy-dp.hpp"
template <class T>
pair<vector<long long>, vector<T>> LucyDP(long long n, function<T(ll)> point_value, function<T(ll)> prefix_sum) {
using ll = long long;
assert(n >= 1);
const ll sq = Math::isqrt(n);
vector<ll> qs(sq * 2 - (n / sq == sq));
for (int i = 0; i < sq; i++) qs[i] = i + 1;
for (int i = 0; i < sq; i++) qs[qs.size() - 1 - i] = n / (i + 1);
vector<T> s(qs.size());
auto v1 = point_value(1);
for (int i = 0; i < (int)qs.size(); i++) s[i] = prefix_sum(qs[i]) - v1;
auto ps = PrimeSieve::table(sq);
for (ll p : ps) {
auto v = point_value(p);
for (int i = (int)qs.size() - 1; i >= 0; i--) {
ll q = qs[i];
if (p * p > q) break;
ll x = q / p;
int j = x <= sq ? x - 1 : (int)s.size() - n / x;
s[i] -= (s[j] - s[p - 2]) * v;
}
}
return {qs, s};
}
/**
* @brief Lucy DP
* @docs docs/number-theory/lucy-dp.md
*/
#line 2 "number-theory/sum-of-multiplicative-function.hpp"
#line 5 "number-theory/sum-of-multiplicative-function.hpp"
template <class T>
T SumOfMultiplicativeFunction(long long n, T e, vector<T> prime_sum, function<T(long long, int, long long)> prime_power_value) {
using ll = long long;
ll sq = Math::isqrt(n);
auto ps = PrimeSieve::table(sq);
auto dfs = [&](auto dfs, ll x, T f, int p_idx, int r, ll q) -> T {
ll m = n / x;
T ret{};
ll p = ps[p_idx];
ret += f * prime_power_value(p, r + 1, q * p);
if (p * p <= m) ret += dfs(dfs, x * p, f, p_idx, r + 1, q * p);
f *= prime_power_value(p, r, q);
ret += f * (prime_sum[m <= sq ? m - 1 : prime_sum.size() - x] - prime_sum[p - 1]);
for (int j = p_idx + 1; j < (int)ps.size(); j++) {
ll p1 = ps[j];
if (p1 * p1 > m) break;
ret += dfs(dfs, x * p1, f, j, 1, p1);
}
return ret;
};
T ret = prime_sum.back() + e;
for (int i = 0; i < (int)ps.size(); i++) {
int p = ps[i];
ret += dfs(dfs, p, e, i, 1, p);
}
return ret;
}
#line 9 "verify/number-theory/LC_sum_of_multiplicative_function.test.cpp"
int main() {
int T;
in(T);
const mint inv2 = mint(2).inv();
while (T--) {
ll N;
mint a, b;
in(N, a, b);
auto sum0 = LucyDP<mint>(N, [&](ll p) { return 1; }, [&](ll p) { return p; }).second;
auto sum1 = LucyDP<mint>(N, [&](ll p) { return p; }, [&](ll p) { return mint(p) * (p + 1) * inv2; }).second;
vector<mint> sum(sum0.size());
rep(i, 0, sum.size()) sum[i] = sum0[i] * a + sum1[i] * b;
mint ans = SumOfMultiplicativeFunction<mint>(N, 1, sum, [&](ll p, int e, ll q) { return a * e + b * p; });
out(ans);
}
}
verify/number-theory/LC_sum_of_multiplicative_function.test.cpp
math/util.hpp