#define PROBLEM "https://yukicoder.me/problems/no/2613" //#include "../default/t.cpp"; //#include "../modint/MontgomeryModInt.cpp" //#include "../poly/NTTmint.cpp" //#include "../numtheory/fastFactorize.cpp" //#include "../numtheory/primitiveRoot.cpp" //#include "../poly/mulConvolution.cpp" #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #define INT128_MAX (__int128)(((unsigned __int128) 1 << ((sizeof(__int128) * __CHAR_BIT__) - 1)) - 1) #define INT128_MIN (-INT128_MAX - 1) #define clock chrono::steady_clock::now().time_since_epoch().count() namespace R = std::ranges; namespace V = std::views; using namespace std; using ll = long long; using ull = unsigned long long; using ldb = long double; using pii = pair; using pll = pair; template ostream& operator<<(ostream& os, const pair pr) { return os << pr.first << ' ' << pr.second; } template ostream& operator<<(ostream& os, const array &arr) { for(const T &X : arr) os << X << ' '; return os; } template ostream& operator<<(ostream& os, const vector &vec) { for(const T &X : vec) os << X << ' '; return os; } template ostream& operator<<(ostream& os, const set &s) { for(const T &x : s) os << x << ' '; return os; } //reference: https://github.com/NyaanNyaan/library/blob/master/modint/montgomery-modint.hpp#L10 //note: mod should be a prime less than 2^30. template struct MontgomeryModInt { using mint = MontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 res = 1, base = mod; for(i32 i = 0; i < 31; i++) res *= base, base *= base; return -res; } static constexpr u32 get_mod() { return mod; } static constexpr u32 n2 = -u64(mod) % mod; //2^64 % mod static constexpr u32 r = get_r(); //-P^{-1} % 2^32 u32 a; static u32 reduce(const u64 &b) { return (b + u64(u32(b) * r) * mod) >> 32; } static u32 transform(const u64 &b) { return reduce(u64(b) * n2); } MontgomeryModInt() : a(0) {} MontgomeryModInt(const int64_t &b) : a(transform(b % mod + mod)) {} mint pow(u64 k) const { mint res(1), base(*this); while(k) { if (k & 1) res *= base; base *= base, k >>= 1; } return res; } mint inverse() const { return (*this).pow(mod - 2); } u32 get() const { u32 res = reduce(a); return res >= mod ? res - mod : res; } mint& operator+=(const mint &b) { if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } mint& operator-=(const mint &b) { if (i32(a -= b.a) < 0) a += 2 * mod; return *this; } mint& operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } mint& operator/=(const mint &b) { a = reduce(u64(a) * b.inverse().a); return *this; } mint operator-() { return mint() - mint(*this); } bool operator==(mint b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } bool operator!=(mint b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } friend mint operator+(mint a, mint b) { return a += b; } friend mint operator-(mint a, mint b) { return a -= b; } friend mint operator*(mint a, mint b) { return a *= b; } friend mint operator/(mint a, mint b) { return a /= b; } friend ostream& operator<<(ostream& os, const mint& b) { return os << b.get(); } friend istream& operator>>(istream& is, mint& b) { int64_t val; is >> val; b = mint(val); return is; } }; using mint = MontgomeryModInt<998244353>; //reference: https://judge.yosupo.jp/submission/69896 //remark: MOD = 2^K * C + 1, R is a primitive root modulo MOD //remark: a.size() <= 2^K must be satisfied //some common modulo: 998244353 = 2^23 * 119 + 1, R = 3 // 469762049 = 2^26 * 7 + 1, R = 3 // 1224736769 = 2^24 * 73 + 1, R = 3 template> struct NTT { using u32 = uint32_t; static constexpr u32 mod = (1 << k) * c + 1; static constexpr u32 get_mod() { return mod; } static void ntt(vector &a, bool inverse) { static array w, w_inv; if (w[0] == 0) { Mint root = 2; while(root.pow((mod - 1) / 2) == 1) root += 1; for(int i = 0; i < 30; i++) w[i] = -(root.pow((mod - 1) >> (i + 2))), w_inv[i] = 1 / w[i]; } int n = ssize(a); if (not inverse) { for(int m = n; m >>= 1; ) { Mint ww = 1; for(int s = 0, l = 0; s < n; s += 2 * m) { for(int i = s, j = s + m; i < s + m; i++, j++) { Mint x = a[i], y = a[j] * ww; a[i] = x + y, a[j] = x - y; } ww *= w[__builtin_ctz(++l)]; } } } else { for(int m = 1; m < n; m *= 2) { Mint ww = 1; for(int s = 0, l = 0; s < n; s += 2 * m) { for(int i = s, j = s + m; i < s + m; i++, j++) { Mint x = a[i], y = a[j]; a[i] = x + y, a[j] = (x - y) * ww; } ww *= w_inv[__builtin_ctz(++l)]; } } Mint inv = 1 / Mint(n); for(Mint &x : a) x *= inv; } } static vector conv(vector a, vector b) { int sz = ssize(a) + ssize(b) - 1; int n = bit_ceil((u32)sz); a.resize(n, 0); ntt(a, false); b.resize(n, 0); ntt(b, false); for(int i = 0; i < n; i++) a[i] *= b[i]; ntt(a, true); a.resize(sz); return a; } }; //#include "poly/NTTmint.cpp" //#include "modint/MontgomeryModInt.cpp" struct mulConvolution { const int P, root; vector powR, logR; int primitiveRoot(int p) { vector pf; { int tmp = p - 1; for(int i = 2; i * i <= (p - 1); i++) { if (tmp % i != 0) continue; pf.emplace_back(i); while(tmp % i == 0) tmp /= i; } if (tmp != 1) pf.emplace_back(tmp); } auto modPow = [p](ll a, int x) -> int { if (x == 0) return 1; if (a == 0) return 0; ll b = 1; while(x) { if (x & 1) b = b * a % p; a = a * a % p, x >>= 1; } return b; }; for(int r = 1; ; r++) { bool isRoot = true; for(int d : pf) { if (modPow(r, (p - 1) / d) == 1) { isRoot = false; break; } } if (isRoot) return r; } } mulConvolution(int _P) : P(_P), root(primitiveRoot(_P)), powR(P - 1), logR(P, -1) { for(int i = 0, tmp = 1; i < P - 1; i++, tmp = (ll)tmp * root % P) powR[i] = tmp, logR[tmp] = i; } template vector transform(vector &f) { assert(ssize(f) == P); vector g(P - 1); for(int i = 1; i < P; i++) g[logR[i]] = f[i]; return g; } template vector invTransform(vector &f) { assert(ssize(f) == P - 1); vector g(P); for(int i = 0; i < P - 1; i++) g[powR[i]] = f[i]; return g; } template vector mulConv(vector a, vector b, vector(*conv)(vector, vector)) { Mint zero = accumulate(a.begin(), a.end(), mint(0)) * b[0] + accumulate(b.begin() + 1, b.end(), mint(0)) * a[0]; a = transform(a), b = transform(b); a = conv(a, b); for(int i = P - 1; i < 2 * P - 3; i++) a[i - (P - 1)] += a[i]; a.resize(P - 1); a = invTransform(a); a[0] = zero; return a; } }; int p; int fac[200000], facInv[200000]; int C(int a, int b) { if (b > a or b < 0) return 0; else return (ll)fac[a] * facInv[b] % p * facInv[a - b] % p; } NTT ntt; signed main() { ios::sync_with_stdio(false), cin.tie(NULL); ll n; cin >> n >> p; fac[0] = 1; for(int i = 1; i < p; i++) fac[i] = (ll)fac[i - 1] * i % p; facInv[p - 1] = 1; for(int i = 0; i < p - 2; i++) facInv[p - 1] = (ll)facInv[p - 1] * fac[p - 1] % p; for(int i = p - 2; i >= 0; i--) facInv[i] = (ll)facInv[i + 1] * (i + 1) % p; mulConvolution mu(p); vector f(p); f[1] = 1; while(n) { int nd = n % p; n /= p; vector g(p); for(int i = 0; i < p; i++) g[C(nd, i)] += 1; f = mu.mulConv(f, g, ntt.conv); } mint ans = 0; for(int i = 1; i < p; i++) ans += f[i] * i; cout << ans << '\n'; return 0; }