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Author: BrandonTang89

Other_Tasks/CF2060_F_Multiplicative_Arrays.cpp

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+#pragma GCC optimize("Ofast")
+#pragma GCC optimize("unroll-loops")
+#include <bits/stdc++.h>  
+using namespace std;
+
+typedef long long ll;
+typedef vector<int> vi;
+#define fast_cin() ios_base::sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL);
+
+ll n, k;
+ll logk;
+constexpr ll MOD = 998244353;
+
+int _sieve_size;
+constexpr ll inf = 1e18;
+vector<ll> lowestpf, primes;
+void smallestPF(ll upperbound = (ll) 1e5+2){
+    _sieve_size = upperbound + 1;
+    lowestpf.assign(_sieve_size, inf);
+
+    for(ll i = 2; i < _sieve_size; i++){
+        if(lowestpf[i] == inf){
+            lowestpf[i] = i;
+            for(ll j = i * i; j < _sieve_size; j += i){
+                lowestpf[j] = min(lowestpf[j], i);
+            }
+            primes.push_back(i);
+        }
+    }
+} // Time: O(n log log n)
+
+constexpr ll mod(ll a, ll n) { return (a % n + n) % n; }
+constexpr ll extEuclid(ll a, ll b, ll &x, ll &y) {  // pass x and y by ref
+    ll xx = y = 0;
+    ll yy = x = 1;
+    while (b) {  // repeats until b == 0
+        ll q = a / b;
+        tie(a, b) = tuple(b, a % b);
+        tie(x, xx) = tuple(xx, x - q * xx);
+        tie(y, yy) = tuple(yy, y - q * yy);
+    }
+    return a;  // returns gcd(a, b)
+	// ax + by = gcd(a, b)
+}
+constexpr ll modInverse(ll a, ll n = MOD) {  // returns modular inverse of a mod n
+    ll x, y;
+    extEuclid(a, n, x, y);
+    return mod(x, n);
+}
+
+
+ll memo_comb[50][50];
+ll comb(int nn, int r){
+    assert(nn >= 0 && r >= 0);
+    if (r == 0 || nn == r) return 1;
+    if (nn < r) return 0;
+    if (memo_comb[nn][r] != -1) return memo_comb[nn][r];
+    return memo_comb[nn][r] =  (comb(nn - 1, r - 1) + comb(nn - 1, r)) % MOD;
+}
+
+ll stars(int stars, int bins){
+    return comb(stars + bins - 1, bins - 1);
+}
+
+vector<vector<ll>> f;
+
+int main(){
+    smallestPF();
+    memset(memo_comb, -1, sizeof memo_comb);
+    int tc;
+    cin >> tc;
+    while(tc--){
+        cin >> k >> n;
+        logk = 1;
+        while ((1LL << logk) <= k) logk++;
+
+        f.assign(k+2, vector<ll>(logk+2, 0));
+        for (int x=2; x<=k; x++){
+            int p = lowestpf[x];
+            int y = x;
+            int q = 0;
+            while (y % p == 0){
+                y /= p;
+                q++;
+            }
+
+            if (y == 1){
+                for (int a=1; a<=logk; a++){
+                    f[x][a] = stars(q-a, a);
+                }
+            }
+            else{
+                for (int a=1; a<=logk; a++){
+                    for (int fresh=0; fresh<a; fresh++){
+                        f[x][a] += ((comb(a, fresh) * f[y][a-fresh]) % MOD) * stars(q-fresh, a) % MOD;
+                        f[x][a] %= MOD;
+                    }
+                }
+            }
+        }
+
+        vector<ll> np1choose(logk+5, 0);
+        for (ll a=1; a<np1choose.size(); a++){
+            ll num = 1;
+            for (int i=0; i<a; i++){
+                num = (num * (n+1-i)) % MOD;
+            }
+
+            ll den = 1;
+            for (int i=1; i<=a; i++){
+                den = (den * i) % MOD;
+            }
+
+            ll deninv = modInverse(den);
+            np1choose[a] = (num * deninv) % MOD;
+        }
+
+        cout << n << " "; // 1 is handled separately
+
+        for (int x =2; x<=k; x++){
+            ll ans = 0;
+            for (int i=1; i<=min(n, logk); i++){
+                ans += (f[x][i] * np1choose[i+1]) % MOD;
+                ans %= MOD;
+            }
+            cout << ans << " ";
+        }
+        cout << endl;
+    }
+
+
+    return 0;
+}