結果
| 問題 |
No.215 素数サイコロと合成数サイコロ (3-Hard)
|
| ユーザー |
 shinchan
|
| 提出日時 | 2025-04-29 23:58:02 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 1,306 ms / 4,000 ms |
| コード長 | 21,597 bytes |
| コンパイル時間 | 3,894 ms |
| コンパイル使用メモリ | 238,204 KB |
| 実行使用メモリ | 22,844 KB |
| 最終ジャッジ日時 | 2025-04-29 23:58:11 |
| 合計ジャッジ時間 | 8,493 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge4 |
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| ファイルパターン | 結果 |
|---|---|
| other | AC * 2 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
#define all(v) (v).begin(),(v).end()
#define pb(a) push_back(a)
#define rep(i, n) for(int i=0;i<n;i++)
#define foa(e, v) for(auto&& e : v)
using ll = long long;
const ll MOD7 = 1000000007, MOD998 = 998244353, INF = (1LL << 60);
#define dout(a) cout<<fixed<<setprecision(10)<<a<<endl;
template<int MOD> struct Modint {
long long val;
constexpr Modint(long long v = 0) noexcept : val(v % MOD) { if (val < 0) val += MOD; }
constexpr int mod() const { return MOD; }
constexpr long long value() const { return val; }
constexpr Modint operator - () const noexcept { return val ? MOD - val : 0; }
constexpr Modint operator + (const Modint& r) const noexcept { return Modint(*this) += r; }
constexpr Modint operator - (const Modint& r) const noexcept { return Modint(*this) -= r; }
constexpr Modint operator * (const Modint& r) const noexcept { return Modint(*this) *= r; }
constexpr Modint operator / (const Modint& r) const noexcept { return Modint(*this) /= r; }
constexpr Modint& operator += (const Modint& r) noexcept {
val += r.val;
if (val >= MOD) val -= MOD;
return *this;
}
constexpr Modint& operator -= (const Modint& r) noexcept {
val -= r.val;
if (val < 0) val += MOD;
return *this;
}
constexpr Modint& operator *= (const Modint& r) noexcept {
val = val * r.val % MOD;
return *this;
}
constexpr Modint& operator /= (const Modint& r) noexcept {
long long a = r.val, b = MOD, u = 1, v = 0;
while (b) {
long long t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
val = val * u % MOD;
if (val < 0) val += MOD;
return *this;
}
constexpr bool operator == (const Modint& r) const noexcept { return this->val == r.val; }
constexpr bool operator != (const Modint& r) const noexcept { return this->val != r.val; }
friend constexpr istream& operator >> (istream& is, Modint<MOD>& x) noexcept {
is >> x.val;
x.val %= MOD;
if (x.val < 0) x.val += MOD;
return is;
}
friend constexpr ostream& operator << (ostream& os, const Modint<MOD>& x) noexcept {
return os << x.val;
}
constexpr Modint<MOD> pow(long long n) noexcept {
if (n == 0) return 1;
if (n < 0) return this->pow(-n).inv();
Modint<MOD> ret = pow(n >> 1);
ret *= ret;
if (n & 1) ret *= *this;
return ret;
}
constexpr Modint<MOD> inv() const noexcept {
long long a = this->val, b = MOD, u = 1, v = 0;
while (b) {
long long t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
return Modint<MOD>(u);
}
};
const int MOD = MOD7;
using mint = Modint<MOD>;
long long modinv(long long a, long long MOD) {
long long b = MOD, u = 1, v = 0;
while (b) {
long long t = a / b;
a -= t * b; std::swap(a, b);
u -= t * v; std::swap(u, v);
}
u %= MOD;
if (u < 0) u += MOD;
return u;
}
long long modpow(long long a, long long n, long long MOD) {
long long res = 1;
a %= MOD;
if(n < 0) {
n = -n;
a = modinv(a, MOD);
}
while (n > 0) {
if (n & 1) res = res * a % MOD;
a = a * a % MOD;
n >>= 1;
}
return res;
}
namespace NTT {
int calc_primitive_root(int MOD) {
if (MOD == 2) return 1;
if (MOD == 167772161) return 3;
if (MOD == 469762049) return 3;
if (MOD == 754974721) return 11;
if (MOD == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
long long x = (MOD - 1) >> 1;
while (x % 2 == 0) x >>= 1;
for (long long i = 3; i * i <= x; i += 2) {
if (x % i == 0) {
divs[cnt ++] = i;
while (x % i == 0) x /= i;
}
}
if (x > 1) divs[cnt++] = x;
for (int g = 2;; ++ g) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (modpow(g, (MOD - 1) / divs[i], MOD) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
int get_fft_size(int N, int M) {
int size_a = 1, size_b = 1;
while (size_a < N) size_a <<= 1;
while (size_b < M) size_b <<= 1;
return std::max(size_a, size_b) << 1;
}
template<class mint> void trans(std::vector<mint>& v, bool inv = false) {
if (v.empty()) return;
int N = (int) v.size();
int MOD = v[0].mod();
int PR = calc_primitive_root(MOD);
static bool first = true;
static std::vector<long long> vbw(30), vibw(30);
if (first) {
first = false;
for (int k = 0; k < 30; ++ k) {
vbw[k] = modpow(PR, (MOD - 1) >> (k + 1), MOD);
vibw[k] = modinv(vbw[k], MOD);
}
}
for (int i = 0, j = 1; j < N - 1; ++ j) {
for (int k = N >> 1; k > (i ^= k); k >>= 1);
if (i > j) std::swap(v[i], v[j]);
}
for (int k = 0, t = 2; t <= N; ++ k, t <<= 1) {
long long bw = vbw[k];
if (inv) bw = vibw[k];
for (int i = 0; i < N; i += t) {
mint w = 1;
for (int j = 0; j < (t >> 1); ++ j) {
int j1 = i + j, j2 = i + j + (t >> 1);
mint c1 = v[j1], c2 = v[j2] * w;
v[j1] = c1 + c2;
v[j2] = c1 - c2;
w *= bw;
}
}
}
if (inv) {
long long invN = modinv(N, MOD);
for (int i = 0; i < N; ++ i) v[i] = v[i] * invN;
}
}
static constexpr int MOD0 = 754974721;
static constexpr int MOD1 = 167772161;
static constexpr int MOD2 = 469762049;
using mint0 = Modint<MOD0>;
using mint1 = Modint<MOD1>;
using mint2 = Modint<MOD2>;
static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1);
static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2);
static const mint2 imod01 = 187290749; // imod1 / MOD0;
// small case (T = mint, long long)
template<class T> std::vector<T> naive_mul
(const std::vector<T>& A, const std::vector<T>& B) {
if (A.empty() || B.empty()) return {};
int N = (int) A.size(), M = (int) B.size();
std::vector<T> res(N + M - 1);
for (int i = 0; i < N; ++ i)
for (int j = 0; j < M; ++ j)
res[i + j] += A[i] * B[j];
return res;
}
};
// mint
template<class mint> std::vector<mint> convolution
(const std::vector<mint>& A, const std::vector<mint>& B) {
if (A.empty() || B.empty()) return {};
int N = (int) A.size(), M = (int) B.size();
if (std::min(N, M) < 30) return NTT::naive_mul(A, B);
int MOD = A[0].mod();
int size_fft = NTT::get_fft_size(N, M);
if (MOD == 998244353) {
std::vector<mint> a(size_fft), b(size_fft), c(size_fft);
for (int i = 0; i < N; ++i) a[i] = A[i];
for (int i = 0; i < M; ++i) b[i] = B[i];
NTT::trans(a), NTT::trans(b);
std::vector<mint> res(size_fft);
for (int i = 0; i < size_fft; ++i) res[i] = a[i] * b[i];
NTT::trans(res, true);
res.resize(N + M - 1);
return res;
}
std::vector<NTT::mint0> a0(size_fft, 0), b0(size_fft, 0), c0(size_fft, 0);
std::vector<NTT::mint1> a1(size_fft, 0), b1(size_fft, 0), c1(size_fft, 0);
std::vector<NTT::mint2> a2(size_fft, 0), b2(size_fft, 0), c2(size_fft, 0);
for (int i = 0; i < N; ++ i) {
a0[i] = A[i].value();
a1[i] = A[i].value();
a2[i] = A[i].value();
}
for (int i = 0; i < M; ++ i) {
b0[i] = B[i].value();
b1[i] = B[i].value();
b2[i] = B[i].value();
}
NTT::trans(a0), NTT::trans(a1), NTT::trans(a2),
NTT::trans(b0), NTT::trans(b1), NTT::trans(b2);
for (int i = 0; i < size_fft; ++i) {
c0[i] = a0[i] * b0[i];
c1[i] = a1[i] * b1[i];
c2[i] = a2[i] * b2[i];
}
NTT::trans(c0, true), NTT::trans(c1, true), NTT::trans(c2, true);
static const mint mod0 = NTT::MOD0, mod01 = mod0 * NTT::MOD1;
std::vector<mint> res(N + M - 1);
for (int i = 0; i < N + M - 1; ++ i) {
int y0 = c0[i].value();
int y1 = (NTT::imod0 * (c1[i] - y0)).value();
int y2 = (NTT::imod01 * (c2[i] - y0) - NTT::imod1 * y1).value();
res[i] = mod01 * y2 + mod0 * y1 + y0;
}
return res;
}
// long long
std::vector<long long> convolution_ll
(const std::vector<long long>& A, const std::vector<long long>& B) {
if (A.empty() || B.empty()) return {};
int N = (int) A.size(), M = (int) B.size();
if (std::min(N, M) < 30) return NTT::naive_mul(A, B);
int size_fft = NTT::get_fft_size(N, M);
std::vector<NTT::mint0> a0(size_fft, 0), b0(size_fft, 0), c0(size_fft, 0);
std::vector<NTT::mint1> a1(size_fft, 0), b1(size_fft, 0), c1(size_fft, 0);
std::vector<NTT::mint2> a2(size_fft, 0), b2(size_fft, 0), c2(size_fft, 0);
for (int i = 0; i < N; ++ i) {
a0[i] = A[i];
a1[i] = A[i];
a2[i] = A[i];
}
for (int i = 0; i < M; ++ i) {
b0[i] = B[i];
b1[i] = B[i];
b2[i] = B[i];
}
NTT::trans(a0), NTT::trans(a1), NTT::trans(a2),
NTT::trans(b0), NTT::trans(b1), NTT::trans(b2);
for (int i = 0; i < size_fft; ++ i) {
c0[i] = a0[i] * b0[i];
c1[i] = a1[i] * b1[i];
c2[i] = a2[i] * b2[i];
}
NTT::trans(c0, true), NTT::trans(c1, true), NTT::trans(c2, true);
static const long long mod0 = NTT::MOD0, mod01 = mod0 * NTT::MOD1;
static const __int128_t mod012 = (__int128_t)mod01 * NTT::MOD2;
std::vector<long long> res(N + M - 1);
for (int i = 0; i < N + M - 1; ++ i) {
int y0 = c0[i].value();
int y1 = (NTT::imod0 * (c1[i] - y0)).value();
int y2 = (NTT::imod01 * (c2[i] - y0) - NTT::imod1 * y1).value();
__int128_t tmp = (__int128_t)mod01 * y2 + (__int128_t)mod0 * y1 + y0;
if(tmp < (mod012 >> 1)) res[i] = tmp;
else res[i] = tmp - mod012;
}
return res;
}
// depends on {modint.cpp}
template <typename mint> struct FPS : std::vector<mint> {
using std::vector<mint>::vector;
// constructor
FPS(const std::vector<mint>& r) : std::vector<mint>(r) {}
// core operator
inline FPS pre(int siz) const {
return FPS(begin(*this), begin(*this) + std::min((int)this->size(), siz));
}
inline FPS rev() const {
FPS res = *this;
reverse(begin(res), end(res));
return res;
}
inline FPS& normalize() {
while (!this->empty() && this->back() == 0) this->pop_back();
return *this;
}
// basic operator
inline FPS operator - () const noexcept {
FPS res = (*this);
for (int i = 0; i < (int)res.size(); ++i) res[i] = -res[i];
return res;
}
inline FPS operator + (const mint& v) const { return FPS(*this) += v; }
inline FPS operator + (const FPS& r) const { return FPS(*this) += r; }
inline FPS operator - (const mint& v) const { return FPS(*this) -= v; }
inline FPS operator - (const FPS& r) const { return FPS(*this) -= r; }
inline FPS operator * (const mint& v) const { return FPS(*this) *= v; }
inline FPS operator * (const FPS& r) const { return FPS(*this) *= r; }
inline FPS operator / (const mint& v) const { return FPS(*this) /= v; }
inline FPS operator << (int x) const { return FPS(*this) <<= x; }
inline FPS operator >> (int x) const { return FPS(*this) >>= x; }
inline FPS& operator += (const mint& v) {
if (this->empty()) this->resize(1);
(*this)[0] += v;
return *this;
}
inline FPS& operator += (const FPS& r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i) (*this)[i] += r[i];
return this->normalize();
}
inline FPS& operator -= (const mint& v) {
if (this->empty()) this->resize(1);
(*this)[0] -= v;
return *this;
}
inline FPS& operator -= (const FPS& r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i) (*this)[i] -= r[i];
return this->normalize();
}
inline FPS& operator *= (const mint& v) {
for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= v;
return *this;
}
inline FPS& operator *= (const FPS& r) {
return *this = convolution((*this), r);
}
inline FPS& operator /= (const mint& v) {
assert(v != 0);
mint iv = v.inv();
for (int i = 0; i < (int)this->size(); ++i) (*this)[i] *= iv;
return *this;
}
inline FPS& operator <<= (int x) {
FPS res(x, 0);
res.insert(res.end(), begin(*this), end(*this));
return *this = res;
}
inline FPS& operator >>= (int x) {
if((int) this->size() <= x) return *this = FPS<mint> (1, 0);
FPS res;
res.insert(res.end(), begin(*this) + x, end(*this));
return *this = res;
}
inline mint eval(const mint& v){
mint res = 0;
for (int i = (int)this->size()-1; i >= 0; --i) {
res *= v;
res += (*this)[i];
}
return res;
}
inline friend FPS gcd(const FPS& f, const FPS& g) {
if (g.empty()) return f;
return gcd(g, f % g);
}
// advanced operation
// df/dx
inline friend FPS diff(const FPS& f) {
int n = (int)f.size();
FPS res(n-1);
for (int i = 1; i < n; ++i) res[i-1] = f[i] * i;
return res;
}
// \int f dx
inline friend FPS intg(const FPS& f) {
int n = (int)f.size();
FPS res(n+1, 0);
for (int i = 0; i < n; ++i) res[i+1] = f[i] / (i+1);
return res;
}
// inv(f), f[0] must not be 0
inline friend FPS inv(const FPS& f, int deg) {
assert(f[0] != 0);
if (deg < 0) deg = (int)f.size();
FPS res({mint(1) / f[0]});
for (int i = 1; i < deg; i <<= 1) {
res = (res + res - res * res * f.pre(i << 1)).pre(i << 1);
}
res.resize(deg);
return res;
}
inline friend FPS inv(const FPS& f) {
return inv(f, f.size());
}
// division, r must be normalized (r.back() must not be 0)
inline FPS& operator /= (const FPS& r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
if (this->size() < r.size()) {
this->clear();
return *this;
}
int need = (int)this->size() - (int)r.size() + 1;
*this = ((*this).rev().pre(need) * inv(r.rev(), need)).pre(need).rev();
return *this;
}
inline FPS& operator %= (const FPS &r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
FPS q = (*this) / r;
return *this -= q * r;
}
inline FPS operator / (const FPS& r) const { return FPS(*this) /= r; }
inline FPS operator % (const FPS& r) const { return FPS(*this) %= r; }
// log(f) = \int f'/f dx, f[0] must be 1
inline friend FPS log(const FPS& f, int deg) {
assert(f[0] == 1);
FPS res = intg(diff(f) * inv(f, deg));
res.resize(deg);
return res;
}
inline friend FPS log(const FPS& f) {
return log(f, f.size());
}
// exp(f), f[0] must be 0
inline friend FPS exp(const FPS& f, int deg) {
assert(f[0] == 0);
FPS res(1, 1);
for (int i = 1; i < deg; i <<= 1) {
res = res * (f.pre(i<<1) - log(res, i<<1) + 1).pre(i<<1);
}
res.resize(deg);
return res;
}
inline friend FPS exp(const FPS& f) {
return exp(f, f.size());
}
// pow(f) = exp(e * log f)
inline friend FPS pow(const FPS& f, long long e, int deg) {
if(e == 0) {
auto ret = FPS(deg, 0);
ret[0] = 1;
return ret;
}
long long i = 0;
while (i < (int)f.size() && f[i] == 0) ++i;
if (i == (int)f.size()) return FPS(deg, 0);
if ((i >= 1 and e >= deg) or i * e >= deg) return FPS(deg, 0);
mint k = f[i];
FPS res = exp(log((f >> i) / k, deg) * e, deg) * k.pow(e) << (e * i);
res.resize(deg);
return res;
}
inline friend FPS pow(const FPS& f, long long e) {
return pow(f, e, f.size());
}
inline friend FPS taylor_shift(FPS f, mint a) {
int n = f.size();
std::vector<mint> fac(n, 1), inv(n, 1), finv(n, 1);
int mod = mint::mod();
for(int i = 2; i < n; i ++) {
fac[i] = fac[i - 1] * i;
inv[i] = -inv[mod % i] * (mod / i);
finv[i] = finv[i - 1] * inv[i];
}
for(int i = 0; i < n; i ++) f[i] *= fac[i];
std::reverse(f.begin(), f.end());
FPS<mint> g(n, 1);
for(int i = 1; i < n; i ++) g[i] = g[i - 1] * a * inv[i];
f = (f * g).pre(n);
std::reverse(f.begin(), f.end());
for(int i = 0; i < n; i ++) f[i] *= finv[i];
return f;
}
};
template <typename mint> FPS<mint> modpow(const FPS<mint> &f, long long n, const FPS<mint> &m) {
if (n == 0) return FPS<mint>(1, 1);
auto t = modpow(f, n / 2, m);
t = (t * t) % m;
if (n & 1) t = (t * f) % m;
return t;
}
vector<mint> BerlekampMassey(const vector<mint> &s) {
const int N = (int)s.size();
vector<mint> b, c;
b.reserve(N + 1);
c.reserve(N + 1);
b.push_back(mint(1));
c.push_back(mint(1));
mint y = mint(1);
for (int ed = 1; ed <= N; ed++) {
int l = int(c.size()), m = int(b.size());
mint x = 0;
for (int i = 0; i < l; i++) x += c[i] * s[ed - l + i];
b.emplace_back(mint(0));
m++;
if (x == mint(0)) continue;
mint freq = x / y;
if (l < m) {
auto tmp = c;
c.insert(begin(c), m - l, mint(0));
for (int i = 0; i < m; i++) c[m - 1 - i] -= freq * b[m - 1 - i];
b = tmp;
y = x;
} else {
for (int i = 0; i < m; i++) c[l - 1 - i] -= freq * b[m - 1 - i];
}
}
reverse(begin(c), end(c));
return c;
}
// Bostan-Mori
// find [x^N] P(x)/Q(x), O(K log K log N)
// deg(Q(x)) = K, deg(P(x)) < K, Q[0] = 1
template <typename mint> mint BostanMori(const FPS<mint> &P, const FPS<mint> &Q, long long N) {
assert(!P.empty() && !Q.empty());
if (N == 0) return P[0] / Q[0];
int qdeg = (int)Q.size();
FPS<mint> P2{P}, minusQ{Q};
P2.resize(qdeg - 1);
for (int i = 1; i < (int)Q.size(); i += 2) minusQ[i] = -minusQ[i];
P2 *= minusQ;
FPS<mint> Q2 = Q * minusQ;
FPS<mint> S(qdeg - 1), T(qdeg);
for (int i = 0; i < (int)S.size(); ++i) {
S[i] = (N % 2 == 0 ? P2[i * 2] : P2[i * 2 + 1]);
}
for (int i = 0; i < (int)T.size(); ++i) {
T[i] = Q2[i * 2];
}
return BostanMori(S, T, N >> 1);
}
// find [x^[[n, n + m)] P(x)/Q(x), O(k log k log n)
template <typename mint> FPS<mint> BostanMori(FPS<mint> P, FPS<mint> Q, long long n, long long m) {
Q.normalize();
int d = Q.size() - 1;
auto add = P / Q;
P -= add * Q;
if(n >= (1LL << 30)) add = FPS<mint>(1, 0);
else add >>= n;
auto rec = [&](auto& rec, FPS<mint> q, ll n) -> FPS<mint> {
if(n <= max(1, d)) {
q.resize(n + d);
auto ret = inv(q);
return FPS<mint>{ret.begin() + n, ret.end()};
}
FPS<mint> minus{q};
for (int i = 1; i < (int)q.size(); i += 2) minus[i] = -minus[i];
auto v2 = minus * q;
FPS<mint> v(d + 1);
for (int i = 0; i < (int)v.size(); i ++) v[i] = v2[i * 2];
int par = (n - d) & 1;
ll nx = (n - d + par) >> 1;
FPS<mint> ret = rec(rec, v, nx);
FPS<mint> ret2(d * 2);
for(int i = 0; i < d; i ++) ret2[i * 2] = ret[i];
auto f = minus * ret2;
return FPS<mint> {f.begin() + (d - par), f.begin() + (d - par) + d};
};
FPS<mint> f = rec(rec, Q, n) * Q;
f.resize(d);
(f *= P) %= Q;
Q.resize(m);
f *= inv(Q);
f += add;
f.resize(m);
return f;
}
vector<mint> calc(vector<ll> to, ll n, ll sz) {
vector dp(n + 1, vector<ll> (sz, 0LL));
dp[0][0] = 1;
foa(e, to) {
for(int i = 0; i < n; i ++) {
for(int j = 0; j + e < sz; j ++) {
dp[i + 1][j + e] += dp[i][j];
if(dp[i + 1][j + e] >= MOD7) dp[i + 1][j + e] -= MOD7;
}
}
}
auto tmp = dp.back();
while(!tmp.empty() and tmp.back() == 0) tmp.pop_back();
vector<mint> ret(tmp.begin(), tmp.end());
return ret;
}
int main() {
cin.tie(0);
ios::sync_with_stdio(false);
ll n, p, c; cin >> n >> p >> c;
vector<ll> v1 = {2, 3, 5, 7, 11, 13};
vector<ll> v2 = {4, 6, 8, 9, 10, 12};
ll N = 1LL << 12;
auto dp1 = calc(v1, p, N);
auto dp2 = calc(v2, c, N);
auto naive = convolution(dp1, dp2);
FPS<mint> a{naive};
N = a.size();
rep(i, N) a[i] = -a[i];
a[0] = 1;
FPS<mint> P(1, 1);
ll Min = max(0LL, n - N);
auto ret = BostanMori(P, a, Min, n - Min);
mint ans = 0;
auto con = convolution(ret, naive);
int sz = con.size();
rep(i, sz) {
if(i + Min >= n) ans += con[i];
}
cout << ans << endl;
return 0;
}