Coding/F_Omega_Numbers.cpp at main · nishantlamba29/Coding · GitHub

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#include <bits/stdc++.h>
using namespace std;
#define int long long
const int mod = 998244353;

const int N = 2e5 + 5;
int spf[N];
int omega[N];

void seive() {
    for(int i=1;i<N;i++) spf[i] = i;
    for(int i=2;i*i<N;i++) {
        if(spf[i] == i) {
            for(int j=i*i;j<N;j+=i) {
                spf[j] = min(spf[j], i);
            }
        }
    }
    for(int i=1;i<N;i++) {
        int cnt = 0, x = i, p = -1;
        while(x != 1) {
            int s = spf[x];
            x /= s;
            if(s != p) cnt++;
            p = s;
        }
        omega[i] = cnt;
    }
}

int modpowr(int a, int b) {
    int res = 1;
    while(b) {
        if(b & 1) res = (res * a) % mod;
        a = (a * a) % mod;
        b >>= 1;
    }
    return res;
}

void solve() {
    int n, k;
    cin >> n >> k;

    vector<int> v(n);
    vector<int> freq(n + 1, 0);
    for(int i=0;i<n;i++) {
        cin >> v[i];
        freq[v[i]]++;
    }

    // cnt_prs[d][s] // count of pairs whose gcd is d and sum of omega values is s

    vector<vector<int>> cnt_prs(n + 1, vector<int>(15, 0));

    for(int i = 1; i <= n; i++) {
        vector<int> cnt(15, 0);
        for(int j=i;j<=n;j+=i) {
            if(freq[j] == 0) continue;
            int w = omega[j];
            cnt[w] += freq[j];
        }
        for(int u=0;u<=6;u++) {
            cnt_prs[i][u+u] += (cnt[u] * (cnt[u] - 1)) / 2;
            for(int v=u+1;v<=6;v++) {
                int sum = u + v; //
                cnt_prs[i][sum] += cnt[u] * cnt[v];
            }
        }
    }

    for(int i=n;i>0;i--) {
        for(int j=2*i;j<=n;j+=i) {
            for(int s=0;s<15;s++) {
                cnt_prs[i][s] -= cnt_prs[j][s];
            }
        }
    }


    // w(x, y) = w(x) + w(y) - w(gcd(x, y))
    // we know w(x) + w(y) = s and gcd(x, y) = d
    // so w(x, y) = s - w(d)
    vector<int> final_cnt(15, 0);

    for(int i=1;i<=n;i++) {
        int wd = omega[i];
        for(int s=wd;s<15;s++) {
            int prs = cnt_prs[i][s];
            final_cnt[s - wd] += prs;
        }
    }

    int ans = 0;
    for(int w = 0; w < 15; w++) {
        ans = (ans + (((final_cnt[w] % mod) * modpowr(w, k)) % mod)) % mod;
    }

    cout << ans << "\n";

}


int32_t main(){
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);
    cout.tie(nullptr);

    seive();

    int _;
    cin >> _;
    while (_-->0) {
        solve();
    }

    return 0;
}