#ifdef NACHIA // #define _GLIBCXX_DEBUG #else #define NDEBUG #endif #include #include #include #include #include #include #include #include #include #include using namespace std; using i64 = long long; using u64 = unsigned long long; #define rep(i,n) for(i64 i=0; i<(i64)(n); i++) #define repr(i,n) for(i64 i=(i64)(n)-1; i>=0; i--) const i64 INF = 1001001001001001001; const char* yn(bool x){ return x ? "Yes" : "No"; } template void chmin(A& l, const A& r){ if(r < l) l = r; } template void chmax(A& l, const A& r){ if(l < r) l = r; } template using nega_queue = priority_queue,greater>; using Modint = atcoder::static_modint<998244353>; #include #include template struct vec; template struct seq_view{ using Ref = typename std::iterator_traits::reference; using Elem = typename std::iterator_traits::value_type; Iter a, b; Iter begin() const { return a; } Iter end() const { return b; } int size() const { return (int)(b-a); } seq_view(Iter first, Iter last) : a(first), b(last) {} seq_view sort() const { std::sort(a, b); return *this; } Ref& operator[](int x){ return *(a+x); } template, class ret = vec> ret sorti(F f = F()) const { ret x(size()); for(int i=0; i> ret col() const { return ret(begin(), end()); } template, class ret = vec>> ret rle(F eq = F()) const { auto x = ret(); for(auto& a : (*this)){ if(x.size() == 0 || !eq(x[x.size()-1].first, a)) x.emp(a, 1); else x[x.size()-1].second++; } return x; } template seq_view sort(F f) const { std::sort(a, b, f); return *this; } Iter uni() const { return std::unique(a, b); } Iter lb(const Elem& x) const { return std::lower_bound(a, b, x); } Iter ub(const Elem& x) const { return std::upper_bound(a, b, x); } int lbi(const Elem& x) const { return lb(x) - a; } int ubi(const Elem& x) const { return ub(x) - a; } seq_view bound(const Elem& l, const Elem& r) const { return { lb(l), lb(r) }; } template Iter lb(const Elem& x, F f) const { return std::lower_bound(a, b, x, f); } template Iter ub(const Elem& x, F f) const { return std::upper_bound(a, b, x, f); } template Iter when_true_to_false(F f) const { if(a == b) return a; return std::lower_bound(a, b, *a, [&](const Elem& x, const Elem&){ return f(x); }); } seq_view same(Elem x) const { return { lb(x), ub(x) }; } template auto map(F f) const { vec r; for(auto& x : *this) r.emp(f(x)); return r; } Iter max() const { return std::max_element(a, b); } Iter min() const { return std::min_element(a, b); } template> Iter min(F f) const { return std::min_element(a, b, f); } seq_view rev() const { std::reverse(a, b); return *this; } }; template struct vec { using Base = typename std::vector; using Iter = typename Base::iterator; using CIter = typename Base::const_iterator; using View = seq_view; using CView = seq_view; vec(){} explicit vec(int n, const Elem& value = Elem()) : a(0 vec(I2 first, I2 last) : a(first, last) {} vec(std::initializer_list il) : a(std::move(il)) {} vec(Base b) : a(std::move(b)) {} operator Base() const { return a; } Iter begin(){ return a.begin(); } CIter begin() const { return a.begin(); } Iter end(){ return a.end(); } CIter end() const { return a.end(); } int size() const { return a.size(); } bool empty() const { return a.empty(); } Elem& back(){ return a.back(); } const Elem& back() const { return a.back(); } vec sortunied(){ vec x = *this; x().sort(); x.a.erase(x().uni(), x.end()); return x; } Iter operator()(int x){ return a.begin() + x; } CIter operator()(int x) const { return a.begin() + x; } View operator()(int l, int r){ return { (*this)(l), (*this)(r) }; } CView operator()(int l, int r) const { return { (*this)(l), (*this)(r) }; } View operator()(){ return (*this)(0,size()); } CView operator()() const { return (*this)(0,size()); } Elem& operator[](int x){ return a[x]; } const Elem& operator[](int x) const { return a[x]; } Base& operator*(){ return a; } const Base& operator*() const { return a; } vec& push(Elem args){ a.push_back(std::move(args)); return *this; } template vec& emp(Args &&... args){ a.emplace_back(std::forward(args) ...); return *this; } template vec& app(Range& x){ for(auto& v : a) emp(v); } Elem pop(){ Elem x = std::move(a.back()); a.pop_back(); return x; } bool operator==(const vec& r) const { return a == r.a; } bool operator!=(const vec& r) const { return a != r.a; } bool operator<(const vec& r) const { return a < r.a; } bool operator<=(const vec& r) const { return a <= r.a; } bool operator>(const vec& r) const { return a > r.a; } bool operator>=(const vec& r) const { return a >= r.a; } vec> pile(int n) const { return vec>(n, *this); } template vec& filter(F f){ int p = 0; for(int q=0; q IStr& operator>>(IStr& is, vec>& v){ for(auto& x:v){ is >> x.first >> x.second; } return is; } template IStr& operator>>(IStr& is, vec& v){ for(auto& x:v){ is >> x; } return is; } template OStr& operator<<(OStr& os, const vec& v){ for(int i=0; i class Comb{ private: std::vector F; std::vector iF; public: void extend(int newN){ int prevN = (int)F.size() - 1; if(prevN >= newN) return; F.resize(newN+1); iF.resize(newN+1); for(int i=prevN+1; i<=newN; i++) F[i] = F[i-1] * Modint::raw(i); iF[newN] = F[newN].inv(); for(int i=newN; i>prevN; i--) iF[i-1] = iF[i] * Modint::raw(i); } Comb(int n = 1){ F.assign(2, Modint(1)); iF.assign(2, Modint(1)); extend(n); } Modint factorial(int n) const { return F[n]; } Modint invFactorial(int n) const { return iF[n]; } Modint invOf(int n) const { return iF[n] * F[n-1]; } Modint comb(int n, int r) const { if(n < 0 || n < r || r < 0) return Modint(0); return F[n] * iF[r] * iF[n-r]; } Modint invComb(int n, int r) const { if(n < 0 || n < r || r < 0) return Modint(0); return iF[n] * F[r] * F[n-r]; } Modint perm(int n, int r) const { if(n < 0 || n < r || r < 0) return Modint(0); return F[n] * iF[n-r]; } Modint invPerm(int n, int r) const { if(n < 0 || n < r || r < 0) return Modint(0); return iF[n] * F[n-r]; } Modint operator()(int n, int r) const { return comb(n,r); } }; } // namespace nachia #include namespace nachia{ bool IsPrime(unsigned long long x) noexcept { if(x <= 1) return false; if(x % 2 == 0) return x == 2; using u64 = unsigned long long; using u128 = __uint128_t; u64 d = x-1; int s = 0; int q = 63; while(!(d&1)){ d >>= 1; s++; } while(!(d >> q)) q--; u64 r = x; for(int t=0; t<6; t++) r*=2-r*x; u128 n2 = -(u128)x % x; auto red = [=](u128 t) noexcept -> u64 { t = (t + (u128)((u64)t*-r)*x) >> 64; return (t >= x) ? t-x : t; }; u64 one = red(n2); for(u64 base : { 2, 325, 9375, 28178, 450775, 9780504, 1795265022 }){ if(base%x==0) continue; u64 a = base = red(base%x*n2); for(int e=q-1; e>=0; e--){ a = red((u128)a*a); if((d>>e)&1) a = red((u128)a*base); } if(a == one) continue; for(int t=1; t> 1) & (~0ull/3)); c = (c & (~0ull/5)) + ((c >> 2) & (~0ull/5)); c = (c & (~0ull/17)) + ((c >> 4) & (~0ull/17)); c = (c * (~0ull/257)) >> 56; return c; #endif } // please ensure x != 0 int MsbIndex(unsigned long long x) noexcept { #ifdef __GNUC__ return 63 - __builtin_clzll(x); #else using u64 = unsigned long long; int q = (x >> 32) ? 32 : 0; auto m = x >> q; constexpr u64 hi = 0x8888'8888; constexpr u64 mi = 0x1111'1111; m = (((m | ~(hi - (m & ~hi))) & hi) * mi) >> 35; m = (((m | ~(hi - (x & ~hi))) & hi) * mi) >> 31; q += (m & 0xf) << 2; q += 0x3333'3333'2222'1100 >> (((x >> q) & 0xf) << 2) & 0xf; return q; #endif } // please ensure x != 0 int LsbIndex(unsigned long long x) noexcept { #ifdef __GNUC__ return __builtin_ctzll(x); #else return MsbIndex(x & -x); #endif } } namespace nachia{ std::vector> Factorize(unsigned long long x){ if(x == 1) return {}; if(IsPrime(x)) return {{x,1}}; using u64 = unsigned long long; using u128 = __uint128_t; u64 X = x; std::vector p; for(u64 i=2; i<100; i+=1+i%2) if(x%i==0){ p.push_back(i); while(x%i==0) x/=i; } u64 r=1; u128 n2=1; auto updX = [&](){ r = x; for(int t=0; t<6; t++) r*=2-r*x; n2 = -(u128)x % x; }; auto red = [&](u128 t) noexcept -> u64 { u64 s = ((u128)x*((u64)t*r)) >> 64; u64 t2 = t >> 64; return t2-s + (t2 < s ? x : 0); }; auto mult = [&](u64 a, u64 b) noexcept { return red((u128)red((u128)a*n2)*b); }; auto gcd = [](u64 a, u64 b) noexcept { if(!a || !b) return a|b; int q = LsbIndex(a|b); b >>= LsbIndex(b); a >>= LsbIndex(a); while(a!=b){ if(a>=LsbIndex(b); } else{ a-=b; a>>=LsbIndex(a); } } return a<>7; v^=v<<17; // Xorshift https://www.jstatsoft.org/article/download/v008i14/916 u64 c = red(v); if(c == 0) continue; auto f = [=](u64 a) noexcept -> u64 { return red((u128)a*a+c); }; u64 a=0, b=f(a); u64 buf = 1, sz = 1, nx = 10; while(true){ while(nx != sz && a != b){ buf = mult(buf, a<=b?b-a:a-b); sz++; a = f(a); b = f(f(b)); } u64 g = gcd(buf, x); if(g != 1){ while(p[pi] % g == 0) p[pi] /= g; p.push_back(g); break; } if(a == b) break; nx = sz * 3 / 2; } } } std::vector> res; for(u64 q : p) if(q != 1){ int e=0; while(X%q == 0){ e++; X/=q; } if(e) res.push_back({ q, e }); } return res; } unsigned long long Totient(unsigned long long x){ auto F = Factorize(x); for(auto f : F) x -= x / f.first; return x; } } // namespace nachia #include #include namespace nachia{ unsigned long long PrimitiveRootDyn(unsigned long long p){ using u64 = unsigned long long; using u128 = __uint128_t; assert(IsPrime(p)); if(p == 2) return 1; auto F = Factorize(p - 1); u64 r = p; for(int t=0; t<6; t++) r*=2-r*p; u64 n2 = -(u128)p % p; auto red = [&](u128 t) noexcept -> u64 { t = (t + (u128)((u64)t*-r)*p) >> 64; return (t >= p) ? t-p : t; }; auto mult = [&](u64 a, u64 b) noexcept { return red((u128)red((u128)a*b)*n2); }; auto powm = [&](u64 a, u64 i) noexcept { u64 b = 1; while(i){ if(i&1){ b=mult(a,b); } a=mult(a,a); i/=2; } return b; }; //static u64 v = 7001; u64 v = 1; while(true){ //v^=v<<13; v^=v>>7; v^=v<<17; // Xorshift https://www.jstatsoft.org/article/download/v008i14/916 v++; u64 vv = v%p; if(vv == 0) continue; bool ok = true; for(auto f : F){ u64 i = (p-1)/f.first; if(powm(vv,i) == 1){ ok = false; break; } } if(ok) break; } return v%p; } } // namespace nachia void testcase(){ i64 N, P; cin >> N >> P; auto cycomb = [&](const vector& a, const vector& b){ auto res = atcoder::convolution(a, b); rep(i,P-2) res[i] += res[P-1+i]; res.resize(P-1); return res; }; using dynMod = atcoder::modint; dynMod::set_mod(P); auto comb = nachia::Comb(P-1); i64 g = nachia::PrimitiveRootDyn(P); vector discreteLog(P); for(i64 i=0, p=1; i res(P-1); res[0] = 1; while(N){ i64 n = N % P; vector A(P-1); rep(i,n+1) A[discreteLog[comb(n, i).val()]] += 1; res = cycomb(res, A); N /= P; } Modint ans = 0; rep(i,P) if(i) ans += res[discreteLog[i]] * i; cout << ans.val() << endl; } int main(){ ios::sync_with_stdio(false); cin.tie(nullptr); #ifdef NACHIA int T; cin >> T; for(int t=0; t