#define AUTO_GENERATED
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/sum_of_totient_function
#include "../../../src/nt/mfsum/min25/lib.hpp"
using namespace tifa_libs;
CEXP u32 MOD = 998244353;
#include "../../../src/math/ds/mint/bs/lib.hpp"
using namespace tifa_libs;
using mint = mint_bs<MOD>;
mint f(u64 p, u64 c) {
u64 res = 1;
while (--c) res = res * p;
return res * (p - 1);
}
int main() {
std::cin.tie(nullptr)->std::ios::sync_with_stdio(false);
u64 n;
std::cin >> n;
tifa_libs::min25_sieve<mint, f> min25(n);
auto h0 = min25.sum_pk(0), h1 = min25.sum_pk(1);
flt_ (u32, i, 0, (u32)h1.size()) h1[i] -= h0[i];
std::cout << min25.run(h1) << '\n';
return 0;
}
#line 1 "test/cpv/library-checker-number_theory/sum_of_totient_function.min25.mints-bs.cpp"
#define AUTO_GENERATED
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/sum_of_totient_function
#line 2 "src/nt/mfsum/min25/lib.hpp"
#line 2 "src/math/div64/lib.hpp"
#line 2 "src/util/alias/num/lib.hpp"
#line 2 "src/util/util/lib.hpp"
// https://github.com/Tiphereth-A/CP-lib
#include <bits/extc++.h>
// clang-format off
namespace tifa_libs {
#define CEXP constexpr
#define CEXPE constexpr explicit
#define CR const&
#define CP const*
#define PC *const
#define CPC const*const
#define TPN typename
#define NE noexcept
#define CNE const noexcept
#define ND [[nodiscard]]
#define cT_(...) std::conditional_t<sizeof(__VA_ARGS__) <= sizeof(size_t) * 2, __VA_ARGS__, __VA_ARGS__ CR>
// NOLINTNEXTLINE(misc-const-correctness)
#define flt_(T, i, l, r, ...) for (T i = (l), i##e = (r)__VA_OPT__(, ) __VA_ARGS__; i < i##e; ++i)
#define retif_(cond, if_true, ...) if cond return if_true __VA_OPT__(; else return __VA_ARGS__)
#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif
using namespace std::ranges;
using namespace std::literals;
template <class T>
CEXP T abs(T x) NE { retif_((x < 0), -x, x); }
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/alias/num/lib.hpp"
// clang-format off
namespace tifa_libs {
#define mk0_(w, t) using w = t; using c##w = const t
#define mk_(w, t) mk0_(w, t); CEXP w operator""_##w(unsigned long long x) NE { return (w)x; }
mk_(i8, int8_t) mk_(u8, uint8_t) mk_(i16, int16_t) mk_(u16, uint16_t) mk_(i32, int32_t) mk_(u32, uint32_t) mk_(i64, int64_t) mk_(u64, uint64_t) mk_(isz, ssize_t) mk_(usz, size_t) mk_(chr, char) mk_(schr, signed char) mk_(uchr, unsigned char) mk_(sint, signed) mk_(uint, unsigned);
mk0_(i128, __int128_t); mk0_(u128, __uint128_t); mk0_(f32, float); mk0_(f64, double); mk0_(f128, long double);
#undef mk0_
#undef mk_
} // namespace tifa_libs
// clang-format on
#line 4 "src/math/div64/lib.hpp"
namespace tifa_libs {
CEXP i64 div_i64d(i64 a, i64 b) NE { return i64(f64(a) / f64(b)); }
CEXP u64 div_u64d(u64 a, u64 b) NE { return u64(f64(a) / f64(b)); }
CEXP i64 div_i64(i64 a, i64 b) NE { retif_((a <= 1'000'000'000'000), div_i64d(a, b), a / b); }
CEXP u64 div_u64(u64 a, u64 b) NE { retif_((a <= 1'000'000'000'000), div_u64d(a, b), a / b); }
} // namespace tifa_libs
#line 2 "src/math/qpow/basic/lib.hpp"
#line 4 "src/math/qpow/basic/lib.hpp"
namespace tifa_libs {
template <class T>
CEXP T qpow(T a, u64 b, cT_(T) init_v = T{1}) NE {
T res = init_v;
for (; b; b >>= 1, a = a * a) {
while (!(b & 1)) b >>= 1, a = a * a;
res = res * a;
}
return res;
}
} // namespace tifa_libs
#line 2 "src/math/sum_ik_flist/lib.hpp"
#line 4 "src/math/sum_ik_flist/lib.hpp"
namespace tifa_libs {
template <class T>
CEXP T sum_i0(T n) NE { return n; }
template <class T>
CEXP T sum_i1(T n) NE { return n * (n + 1) / 2; }
template <class T>
CEXP T sum_i2(T n) NE { return sum_i1(n) * (n * 2 + 1) / 3; }
template <class T>
CEXP T sum_i3(T n) NE {
const auto _ = sum_i1(n);
return _ * _;
}
template <class T>
CEXP T sum_i4(T n) NE { return sum_i2(n) * (sum_i1(n) * 6 - 1) / 5; }
template <class T>
CEXP T sum_i5(T n) NE { return sum_i3(n) * (sum_i1(n) * 4 - 1) / 3; }
template <class T>
CEXP T sum_i6(T n) NE {
const auto _ = sum_i1(n);
return sum_i2(n) * (_ * (_ * 2 - 1) * 6 + 1) / 7;
}
template <class T>
CEXP T sum_i7(T n) NE {
const auto _ = sum_i3(n);
return _ * _ * 2 - sum_i5(n);
}
template <class T>
// NOLINTNEXTLINE(modernize-avoid-c-arrays)
CEXP T (*sum_ik[])(T) NE = {sum_i0<T>, sum_i1<T>, sum_i2<T>, sum_i3<T>, sum_i4<T>, sum_i5<T>, sum_i6<T>, sum_i7<T>};
} // namespace tifa_libs
#line 2 "src/nt/prime_seq/lib.hpp"
#line 2 "src/math/iroot/sqrt/lib.hpp"
#line 2 "src/util/alias/others/lib.hpp"
#line 2 "src/util/consts/lib.hpp"
#line 4 "src/util/consts/lib.hpp"
// clang-format off
namespace tifa_libs {
using std::numbers::pi_v;
template <std::floating_point FP>
inline FP eps_v = std::sqrt(std::numeric_limits<FP>::epsilon());
template <std::floating_point FP>
CEXP void set_eps(FP v) NE { eps_v<FP> = v; }
CEXP u32 TIME = ((__TIME__[0] & 15) << 20) | ((__TIME__[1] & 15) << 16) | ((__TIME__[3] & 15) << 12) | ((__TIME__[4] & 15) << 8) | ((__TIME__[6] & 15) << 4) | (__TIME__[7] & 15);
CEXP auto STR2U16 = [] { std::array<u32, 65536> table{}; table.fill(-1_u32); flt_ (u32, i, 48, 58) flt_ (u32, j, 48, 58) table[i << 8 | j] = (j & 15) * 10 + (i & 15); return table; }();
inline const auto fn_0 = [](auto&&...) NE {};
inline const auto fn_is0 = [](auto x) NE { return x == 0; };
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/alias/others/lib.hpp"
namespace tifa_libs {
template <class T>
struct chash {
CEXP static u64 C = u64(pi_v<f128> * 2e18) | 71;
CEXP u64 operator()(T x) CNE { return __builtin_bswap64(((u64)x ^ TIME) * C); }
};
// clang-format off
#define mk_(w, t) using w = t; using c##w = const t;
mk_(strn, std::string) mk_(strnv, std::string_view)
#undef mk_
template <class T> struct edge_t { T w; u32 u, v; CEXP auto operator<=>(edge_t CR) const = default; }; template <class T> using cedge_t = const edge_t<T>;
template <class T> struct pt3 { T _0, _1, _2; CEXP auto operator<=>(pt3 CR) const = default; }; template <class T> using cpt3 = const pt3<T>;
template <class T> struct pt4 { T _0, _1, _2, _3; CEXP auto operator<=>(pt4 CR) const = default; }; template <class T> using cpt4 = const pt4<T>;
#define mkT_(w, t, ...) template <class T> using w = t __VA_OPT__(, ) __VA_ARGS__; template <class T> using c##w = const t __VA_OPT__(, ) __VA_ARGS__;
mkT_(ptt, std::pair<T, T>) mkT_(alc, std::pmr::polymorphic_allocator<T>) mkT_(vec, std::vector<T>) mkT_(vvec, vec<vec<T>>) mkT_(v3ec, vvec<vec<T>>) mkT_(vecpt, vec<ptt<T>>) mkT_(vvecpt, vvec<ptt<T>>) mkT_(ptvec, ptt<vec<T>>) mkT_(ptvvec, ptt<vvec<T>>)
#undef mkT_
template <class T> using itl = std ::initializer_list<T>;
template <class T, usz ext = std::dynamic_extent> using spn = std::span<T const, ext>;
template <class T, usz N> using arr = std::array<T, N>; template <class T, usz N> using carr = std::array<const T, N>;
template <class U, class T> using vecp = vec<std::pair<U, T>>; template <class U, class T> using vvecp = vvec<std::pair<U, T>>;
template <class U, class T> using vvecp = vvec<std::pair<U, T>>; template <class U, class T> using vvvecp = vvec<vvec<std::pair<U, T>>>;
#ifdef PB_DS_ASSOC_CNTNR_HPP
template <class T, class C = std::less<T>> using set = __gnu_pbds::tree<T, __gnu_pbds::null_type, C>;
template <class K, class V, class C = std::less<K>> using map = __gnu_pbds::tree<K, V, C>;
// hset<u64> s({}, {}, {}, {}, {1<<16});
template <class T, class HF = chash<T>> using hset = __gnu_pbds::gp_hash_table<T, __gnu_pbds::null_type, HF>;
// hmap<u64, int> s({}, {}, {}, {}, {1<<16});
template <class K, class V, class HF = chash<K>> using hmap = __gnu_pbds::gp_hash_table<K, V, HF>;
#else
using std::set, std::map;
template <class T, class HF = chash<T>> using hset = std::unordered_set<T, HF>;
template <class K, class V, class HF = chash<K>> using hmap = std::unordered_map<K, V, HF>;
#endif
#ifdef PB_DS_PRIORITY_QUEUE_HPP
template <class T, class C = std::less<T>> using pq = __gnu_pbds::priority_queue<T, C>;
#else
template <class T, class C = std::less<T>> using pq = std::priority_queue<T, vec<T>, C>;
#endif
template <class T> using pqg = pq<T, std::greater<T>>;
// clang-format on
#define mk1_(V, A, T) using V##A = V<T>;
#define mk_(V, A, T) mk1_(V, A, T) mk1_(c##V, A, T)
#define mk(A, T) mk_(edge_t, A, T) mk_(ptt, A, T) mk_(pt3, A, T) mk_(pt4, A, T) mk_(vec, A, T) mk_(vvec, A, T) mk_(v3ec, A, T) mk_(vecpt, A, T) mk_(vvecpt, A, T) mk_(ptvec, A, T) mk_(ptvvec, A, T) mk1_(spn, A, T) mk1_(itl, A, T)
mk(b, bool) mk(c, chr) mk(i, i32) mk(u, u32) mk(ii, i64) mk(uu, u64) mk(t, isz) mk(z, usz) mk(f, f32) mk(d, f64) mk(s, strn);
#undef mk
#undef mk_
#undef mk1_
} // namespace tifa_libs
#line 4 "src/math/iroot/sqrt/lib.hpp"
namespace tifa_libs {
CEXP u32 isqrt(u64 x) NE {
retif_((!x) [[unlikely]], 0);
#pragma GCC diagnostic ignored "-Wconversion"
csint sh = 31 - (std::bit_width(x) - 1) / 2;
#pragma GCC diagnostic warning "-Wconversion"
u32 u = [](u64 x) NE {
CEXP arr<u8, 192> TAB{128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 144, 145, 146, 147, 148, 149, 150, 151, 151, 152, 153, 154, 155, 156, 156, 157, 158, 159, 160, 160, 161, 162, 163, 164, 164, 165, 166, 167, 167, 168, 169, 170, 170, 171, 172, 173, 173, 174, 175, 176, 176, 177, 178, 179, 179, 180, 181, 181, 182, 183, 183, 184, 185, 186, 186, 187, 188, 188, 189, 190, 190, 191, 192, 192, 193, 194, 194, 195, 196, 196, 197, 198, 198, 199, 200, 200, 201, 201, 202, 203, 203, 204, 205, 205, 206, 206, 207, 208, 208, 209, 210, 210, 211, 211, 212, 213, 213, 214, 214, 215, 216, 216, 217, 217, 218, 219, 219, 220, 220, 221, 221, 222, 223, 223, 224, 224, 225, 225, 226, 227, 227, 228, 228, 229, 229, 230, 230, 231, 232, 232, 233, 233, 234, 234, 235, 235, 236, 237, 237, 238, 238, 239, 239, 240, 240, 241, 241, 242, 242, 243, 243, 244, 244, 245, 246, 246, 247, 247, 248, 248, 249, 249, 250, 250, 251, 251, 252, 252, 253, 253, 254, 254, 255, 255, 255};
u32 u = TAB[(x >> 56) - 64];
u = (u << 7) + (u32)(x >> 41) / u;
return (u << 15) + (u32)((x >> 17) / u);
}(x << 2 * sh);
u >>= sh, u -= (u64)u * u > x;
return u;
}
} // namespace tifa_libs
#line 5 "src/nt/prime_seq/lib.hpp"
namespace tifa_libs {
CEXP vecu prime_seq(u32 n) NE {
vecb sieve(n / 3 + 1, true);
for (u32 p = 5, d = 4, i = 1, sqn = isqrt(n); p <= sqn; p += d = 6 - d, ++i) {
if (!sieve[i]) continue;
for (u64 q = p * p / 3, r = d * p / 3 + (d * p % 3 == 2), s = 2 * p, qe = sieve.size(); q < qe; q += r = s - r) sieve[q] = false;
}
vecu ret{2, 3};
for (u32 p = 5, d = 4, i = 1; p <= n; p += d = 6 - d, ++i)
if (sieve[i]) ret.push_back(p);
while (!ret.empty() && ret.back() > n) ret.pop_back();
return ret;
}
} // namespace tifa_libs
#line 7 "src/nt/mfsum/min25/lib.hpp"
namespace tifa_libs {
// f(p, c) = value of f(p^c)
template <class T, T (*f)(u64, u64)>
class min25_sieve {
u64 m, sqm, s;
vecu p;
ND CEXP u64 idx(u64 n) CNE { retif_((n <= sqm), s - n, div_u64d(m, n)); }
public:
// m^{3/2} in u64
CEXPE min25_sieve(u64 m) NE : m{m}, sqm{isqrt(m)} {
if (assert(m < (1ll << 42)); m) {
u64 hls = div_u64d(m, sqm);
if (hls != 1 && div_u64d(m, hls - 1) == sqm) --hls;
s = hls + sqm, p = prime_seq((u32)sqm);
}
}
ND CEXP vec<T> sum_pk(u32 k) CNE {
auto sik = sum_ik<T>[k];
retif_((!m) [[unlikely]], {});
u64 hls = div_u64d(m, sqm);
if (hls != 1 && div_u64d(m, hls - 1) == sqm) --hls;
vec<T> h(s);
flt_ (u64, i, 1, hls) h[i] = sik(div_u64d(m, i)) - 1;
flt_ (u64, i, 1, sqm + 1) h[s - i] = sik(i) - 1;
for (cu32 x : p) {
T _ = x, pi = h[s - x + 1];
_ = qpow(_, k);
cu64 x2 = u64(x) * x, mx = min(hls, div_u64d(m, x2) + 1);
for (u64 i = 1, ix = x; i < mx; ++i, ix += x) h[i] -= ((ix < hls ? h[ix] : h[s - div_u64d(m, ix)]) - pi) * _;
for (u64 n = sqm; n >= x2; --n) h[s - n] -= (h[s - div_u64d(n, x)] - pi) * _;
}
assert(h.size() == s);
return h;
}
ND CEXP T run(vec<T> fprime) CNE {
retif_((!m) [[unlikely]], {});
assert(fprime.size() == s);
T ans = fprime[idx(m)] + 1;
auto dfs = [&, this](auto&& dfs, u32 i, u32 c, u64 prod, T now) NE -> void {
ans += now * f(p[i], c + 1);
cu64 lim = div_u64d(m, prod);
if (lim >= (u64)p[i] * p[i]) dfs(dfs, i, c + 1, p[i] * prod, now);
now *= f(p[i], c), ans += now * (fprime[idx(lim)] - fprime[idx(p[i])]);
u32 j = i + 1;
for (; j < p.size() && (u64)p[j] * p[j] * p[j] <= lim; ++j) dfs(dfs, j, 1, prod * p[j], now);
for (; j < p.size() && (u64)p[j] * p[j] <= lim; ++j) {
T _ = f(p[j], 2);
u64 i1 = idx(div_u64d(lim, p[j])), i2 = idx(p[j]);
ans += now * (_ += f(p[j], 1) * (fprime[i1] - fprime[i2]));
}
};
flt_ (u32, i, 0, (u32)p.size()) dfs(dfs, i, 1, p[i], 1);
return ans;
}
};
} // namespace tifa_libs
#line 4 "test/cpv/library-checker-number_theory/sum_of_totient_function.min25.mints-bs.cpp"
using namespace tifa_libs;
CEXP u32 MOD = 998244353;
#line 2 "src/math/ds/mint/bs/lib.hpp"
#line 2 "src/nt/mod/barrett/lib.hpp"
#line 4 "src/nt/mod/barrett/lib.hpp"
namespace tifa_libs {
template <u64 MOD, u64 B_ = 1>
struct barrett {
static CEXP u64 B = B_ % MOD, R = ((u128)B << 64) / MOD;
static CEXP u64 reduce(u64 a) NE {
if (u64 q = u64((u128)a * R >> 64); (a = a * B - q * MOD) >= MOD) a -= MOD;
return a;
}
};
template <> // dynamic
struct barrett<0> {
u64 mod, b, r;
CEXP barrett() NE = default;
CEXPE barrett(u64 mod, u64 b = 1) NE { reset(mod, b); }
CEXP void reset(u64 mod_, u64 b_ = 1) NE { assert(mod_), mod = mod_, b = b_ % mod, r = (u64(((u128)b << 64) / mod)); }
ND CEXP u64 reduce(u64 a) CNE {
if (cu64 q = u64((u128)a * r >> 64); (a = a * b - q * mod) >= mod) a -= mod;
return a;
}
};
} // namespace tifa_libs
#line 2 "src/math/ds/mint/_base/lib.hpp"
#line 2 "src/nt/inverse/lib.hpp"
#line 2 "src/nt/gl/inv_gcd/lib.hpp"
#line 2 "src/math/safe_mod/lib.hpp"
#line 2 "src/util/traits/math/lib.hpp"
// clang-format off
#line 4 "src/util/traits/math/lib.hpp"
namespace tifa_libs {
template <class T> concept char_c = std::same_as<T, char> || std::same_as<T, signed char> || std::same_as<T, unsigned char>;
#pragma GCC diagnostic ignored "-Wpedantic"
template <class T> concept s128_c = std::same_as<T, __int128_t> || std::same_as<T, __int128>;
template <class T> concept u128_c = std::same_as<T, __uint128_t> || std::same_as<T, unsigned __int128>;
template <class T> concept i128_c = s128_c<T> || u128_c<T>;
#pragma GCC diagnostic warning "-Wpedantic"
template <class T> concept imost64_c = std::integral<T> && sizeof(T) * __CHAR_BIT__ <= 64;
template <class T> concept smost64_c = imost64_c<T> && std::signed_integral<T>;
template <class T> concept umost64_c = imost64_c<T> && std::unsigned_integral<T>;
template <class T> concept int_c = i128_c<T> || imost64_c<T>;
template <class T> concept sint_c = s128_c<T> || smost64_c<T>;
template <class T> concept uint_c = u128_c<T> || umost64_c<T>;
template <class T> concept arithm_c = std::is_arithmetic_v<T> || int_c<T>;
template <class T> concept mint_c = requires(T x) { {x.mod()} -> uint_c; {x.val()} -> uint_c; };
template <class T> concept dft_c = requires(T x, std::vector<TPN T::data_t> v, u32 n) { {x.size()} -> std::same_as<u32>; x.bzr(n); x.dif(v, n); x.dit(v, n); };
template <class T> concept ntt_c = dft_c<T> && requires(T x) { T::max_size; T::G; };
template <class T> struct to_sint : std::make_signed<T> {};
template <> struct to_sint<u128> { using type = i128; };
template <> struct to_sint<i128> { using type = i128; };
template <class T> using to_sint_t = TPN to_sint<T>::type;
template <class T> struct to_uint : std::make_unsigned<T> {};
template <> struct to_uint<u128> { using type = u128; };
template <> struct to_uint<i128> { using type = u128; };
template <class T> using to_uint_t = TPN to_uint<T>::type;
template <arithm_c T> struct to_bigger : std::make_unsigned<T> {};
#define _(w,ww) template <> struct to_bigger<w> { using type = ww; }
#define _2(w,ww) _(i##w,i##ww); _(u##w,u##ww);
_2(8, 16); _2(16, 32); _2(32, 64); _2(64, 128); _(f32, f64); _(f64, f128);
#undef _2
#undef _
template <class T> using to_bigger_t = TPN to_bigger<T>::type;
template <arithm_c T> CEXP T inf_v = [] {
if CEXP(sint_c<T>) return T(to_uint_t<T>(-1) / 4 - 1);
else if CEXP(uint_c<T>) return T(-1) / 2 - 1;
else return std::numeric_limits<T>::max() / 2 - 1;
}();
} // namespace tifa_libs
// clang-format on
#line 4 "src/math/safe_mod/lib.hpp"
namespace tifa_libs {
template <int_c T>
CEXP T safe_mod(T x, to_uint_t<T> mod) NE {
if CEXP (sint_c<T>) {
if (x <= -(T)mod || x >= (T)mod) x %= (T)mod;
retif_((x < 0), x + (T)mod, x);
} else {
retif_((x >= mod), x % mod, x);
}
}
} // namespace tifa_libs
#line 2 "src/nt/gl/exgcd/lib.hpp"
#line 4 "src/nt/gl/exgcd/lib.hpp"
namespace tifa_libs {
// Binary exgcd
template <uint_c U, bool only_x = false>
CEXP auto exgcd_b(U a, U b) NE {
using T = to_sint_t<U>;
if CEXP (only_x) {
if (!a) return std::make_tuple(b, (T)0);
if (!b) return std::make_tuple(a, (T)1);
} else {
if (!a) return std::make_tuple(b, (T)0, (T) !!b);
if (!b) return std::make_tuple(a, (T)1, (T)0);
}
auto r = std::__countr_zero(a | b);
a >>= r, b >>= r;
T x = (T)a, y = (T)b, s = 1, t = 0, u = 0, v = 1;
while (x) {
while (!(x & 1))
if (x /= 2; !((s | t) & 1)) s /= 2, t /= 2;
else s = (s + (T)b) / 2, t = (t - (T)a) / 2;
while (!(y & 1))
if (y /= 2; !((u | v) & 1)) u /= 2, v /= 2;
else u = (u + (T)b) / 2, v = (v - (T)a) / 2;
if (x >= y) x -= y, s -= u, t -= v;
else y -= x, u -= s, v -= t;
}
if (y > 1) a /= (U)y, b /= (U)y;
if (a && (U)abs(v) >= a) {
const T _ = v / (T)a;
v -= _ * (T)a, u += _ * (T)b;
}
if (b && (U)abs(u) >= b) {
const T _ = u / (T)b;
u -= _ * (T)b, v += _ * (T)a;
}
if (const T u_ = u + (T)b, v_ = v - (T)a; abs(u_) + abs(v_) <= abs(u) + abs(v)) u = u_, v = v_;
if (const T u_ = u - (T)b, v_ = v + (T)a; abs(u_) + abs(v_) <= abs(u) + abs(v)) u = u_, v = v_;
if CEXP (only_x) return std::make_tuple(U(y << r), u);
else return std::make_tuple(U(y << r), u, v);
}
// @return then return tuple(g, x[, y]) s.t. g = gcd(a, b), xa + yb = g, |x| + |y| is the minimal (primary) and x <= y (secondarily)
template <sint_c T, bool only_x = false>
CEXP auto exgcd(T a, T b) NE {
using U = to_uint_t<T>;
if (auto [x, y] = minmax(a, b); x >= 0 && y <= T(U(-1) >> sizeof(U))) return exgcd_b<U, only_x>((U)a, (U)b);
if CEXP (only_x) {
T s = 1, u = 0;
while (b) {
T c = a / b;
std::tie(s, u, a, b) = std::make_tuple(u, s - u * c, b, a - b * c);
}
return std::make_tuple((U)a, s);
} else {
T s = 1, t = 0, u = 0, v = 1;
while (b) {
T c = a / b;
std::tie(s, t, u, v, a, b) = std::make_tuple(u, v, s - u * c, t - v * c, b, a - b * c);
}
return std::make_tuple((U)a, s, t);
}
}
} // namespace tifa_libs
#line 6 "src/nt/gl/inv_gcd/lib.hpp"
namespace tifa_libs {
template <uint_c T>
CEXP ptt<T> inv_gcd(T n, T mod) NE {
using U = to_sint_t<T>;
auto [g, x] = exgcd<U, true>(U(n % mod), (U)mod);
return {g, safe_mod(x, mod)};
}
} // namespace tifa_libs
#line 4 "src/nt/inverse/lib.hpp"
namespace tifa_libs {
// simple but slower: inv(n, mod) -> 1 < n ? mod - inv(mod % n, n) * mod / n : 1;
template <uint_c T, uint_c U>
CEXP U inverse(T n, U mod) NE {
auto [g, x] = inv_gcd(U(n % mod), mod);
assert(g == 1);
return x;
}
} // namespace tifa_libs
#line 2 "src/util/traits/others/lib.hpp"
// clang-format off
#line 4 "src/util/traits/others/lib.hpp"
namespace tifa_libs {
//! only for template without non-type argument
template <class, template <class...> class> CEXP bool specialized_from_v = false;
template <template <class...> class T, class... Args> CEXP bool specialized_from_v<T<Args...>, T> = true;
static_assert(specialized_from_v<vecu, std::vector>);
template <class T> concept container_c = common_range<T> && !std::is_array_v<std::remove_cvref_t<T>> && !std::same_as<std::remove_cvref_t<T>, strn> && !std::same_as<std::remove_cvref_t<T>, strnv>;
template <class T> concept istream_c = std::derived_from<T, std::istream> || std::derived_from<T, std::wistream> || requires(T is) { is.peek(); };
template <class T> concept ostream_c = std::derived_from<T, std::ostream> || std::derived_from<T, std::wostream> || requires(T os) { os.flush(); };
} // namespace tifa_libs
// clang-format on
#line 5 "src/math/ds/mint/_base/lib.hpp"
namespace tifa_libs::mint_impl_ {
struct mint_tag_base {};
template <std::derived_from<mint_tag_base> tag_t>
struct mint : tag_t {
CEXP mint() = default;
CEXP mint(int_c auto v) NE : tag_t(v) {}
using raw_t = tag_t::raw_t;
using sraw_t = to_sint_t<raw_t>;
static CEXP sraw_t smod() NE { return (sraw_t)tag_t::mod(); }
ND CEXP sraw_t sval() CNE { return (sraw_t)tag_t::val(); }
template <int_c T>
CEXPE operator T() CNE { return (T)tag_t::val(); }
CEXP mint& operator+=(mint CR r) NE {
mint::add(r);
return *this;
}
CEXP mint& operator-=(mint CR r) NE {
mint::sub(r);
return *this;
}
CEXP mint& operator*=(mint CR r) NE {
mint::mul(r);
return *this;
}
CEXP mint& operator/=(mint CR r) NE { return *this = *this * r.inv(); }
CEXP mint CR operator+() CNE { return *this; }
CEXP mint operator-() CNE { return tag_t::template neg<mint>(); }
ND CEXP mint inv() CNE { return inverse(tag_t::val(), tag_t::mod()); }
friend CEXP mint operator+(mint l, mint CR r) NE { return l += r; }
friend CEXP mint operator-(mint l, mint CR r) NE { return l -= r; }
friend CEXP mint operator*(mint l, mint CR r) NE { return l *= r; }
friend CEXP mint operator/(mint l, mint CR r) NE { return l /= r; }
friend CEXP bool operator==(mint CR l, mint CR r) NE { return l.val() == r.val(); }
friend CEXP auto operator<=>(mint CR l, mint CR r) NE { return l.sval() <=> r.sval(); }
friend auto& operator>>(istream_c auto& is, mint& x) NE {
i64 _;
is >> _, x = mint(_);
return is;
}
friend auto& operator<<(ostream_c auto& os, mint CR x) NE { return os << x.val(); }
friend CEXP auto abs(mint CR x) NE { return x.val(); }
};
} // namespace tifa_libs::mint_impl_
#line 5 "src/math/ds/mint/bs/lib.hpp"
namespace tifa_libs {
template <u64 MOD_>
class mint_bs_tag : public mint_impl_::mint_tag_base {
static_assert(MOD_ && MOD_ <= UINT32_MAX);
using core = barrett<MOD_>;
public:
static CEXP bool FIXED_MOD = true;
protected:
using raw_t = u32;
raw_t v_{};
CEXP mint_bs_tag() NE = default;
CEXP mint_bs_tag(int_c auto v) NE : v_{mod(v)} {}
public:
static CEXP raw_t mod(sint_c auto v) NE {
if (v >= 0) return mod((to_uint_t<decltype(v)>)v);
if (auto ret = mod((to_uint_t<decltype(v)>)-v); ret) return mod() - ret;
else return ret;
}
static CEXP raw_t mod(uint_c auto v) NE {
if CEXP (umost64_c<decltype(v)>) return (raw_t)core::reduce((u64)v);
else if (v < UINT64_MAX) return (raw_t)core::reduce((u64)v);
else return raw_t(v % mod());
}
static CEXP raw_t mod() NE { return MOD_; }
ND CEXP raw_t val() CNE { return v_; }
CEXP raw_t& data() NE { return v_; }
protected:
template <class mint>
CEXP auto neg() CNE {
mint res;
if (v_) res.v_ = mod() - v_;
return res;
}
CEXP void add(mint_bs_tag CR r) NE {
if ((v_ += r.v_) >= mod()) v_ -= mod();
}
CEXP void sub(mint_bs_tag CR r) NE {
if (i32(v_ -= r.v_) < 0) v_ += mod();
}
CEXP void mul(mint_bs_tag CR r) NE { v_ = (raw_t)core::reduce(u64(v_) * r.v_); }
};
template <u64 MOD>
using mint_bs = mint_impl_::mint<mint_bs_tag<MOD>>;
} // namespace tifa_libs
#line 9 "test/cpv/library-checker-number_theory/sum_of_totient_function.min25.mints-bs.cpp"
using namespace tifa_libs;
using mint = mint_bs<MOD>;
mint f(u64 p, u64 c) {
u64 res = 1;
while (--c) res = res * p;
return res * (p - 1);
}
int main() {
std::cin.tie(nullptr)->std::ios::sync_with_stdio(false);
u64 n;
std::cin >> n;
tifa_libs::min25_sieve<mint, f> min25(n);
auto h0 = min25.sum_pk(0), h1 = min25.sum_pk(1);
flt_ (u32, i, 0, (u32)h1.size()) h1[i] -= h0[i];
std::cout << min25.run(h1) << '\n';
return 0;
}
test/cpv/library-checker-number_theory/sum_of_totient_function.min25.mints-bs.cpp