Number Theory Part 2 Added

Author: zarif98sjs

Algorithm/25 Prime Factorization.cpp

@@ -0,0 +1,190 @@
+
+/** Which of the favors of your Lord will you deny ? **/
+
+#include<bits/stdc++.h>
+using namespace std;
+
+#define LL long long
+#define PII pair<int,int>
+#define PLL pair<LL,LL>
+#define MP make_pair
+#define F first
+#define S second
+#define INF INT_MAX
+
+#define ALL(x) (x).begin(), (x).end()
+#define DBG(x) cerr << __LINE__ << " says: " << #x << " = " << (x) << endl
+#define READ        freopen("alu.txt", "r", stdin)
+#define WRITE       freopen("vorta.txt", "w", stdout)
+
+#include <ext/pb_ds/assoc_container.hpp>
+#include <ext/pb_ds/tree_policy.hpp>
+using namespace __gnu_pbds;
+
+template<class TIn>using indexed_set = tree<TIn, null_type, less<TIn>,rb_tree_tag, tree_order_statistics_node_update>;
+
+/**
+
+PBDS
+-------------------------------------------------
+1) insert(value)
+2) erase(value)
+3) order_of_key(value) // 0 based indexing
+4) *find_by_order(position) // 0 based indexing
+
+**/
+
+template<class T1, class T2>
+ostream &operator <<(ostream &os, pair<T1,T2>&p);
+template <class T>
+ostream &operator <<(ostream &os, vector<T>&v);
+template <class T>
+ostream &operator <<(ostream &os, set<T>&v);
+
+inline void optimizeIO()
+{
+    ios_base::sync_with_stdio(false);
+    cin.tie(NULL);
+}
+
+const int nmax = 2e5+7;
+const LL LINF = 1e17;
+
+template <class T>
+string to_str(T x)
+{
+    stringstream ss;
+    ss<<x;
+    return ss.str();
+}
+
+//bool cmp(const PII &A,const PII &B)
+//{
+//
+//}
+
+/** Bit Sieve **/
+
+const int pnmax = 1e8+7;
+LL LIM;
+bitset<pnmax> bs;   /// can sieve upto 1e8 in ~ 1 sec
+vector<LL> primes;
+
+void bit_sieve(LL upperbound)
+{
+    LIM = upperbound + 1;
+    bs.set();                                     /// set all bits to 1
+    bs[0] = bs[1] = 0;
+    for (LL i = 2; i <= LIM; i++) /** If I don't want to know the primes , it is enough to loop upto sqrt(LIM) here **/
+        if (bs[i])
+        {
+            for (LL j = i * i; j <= LIM; j += i)
+                bs[j] = 0;
+            primes.push_back(i);
+        }
+}
+
+bool isPrime(LL N)
+{
+    if (N <= LIM)
+        return bs[N];                   /// O(1) for small primes
+
+    /** note: only work for N <= (last prime in "primes" vector)^2 . So, if Sieve is done upto 10^6 , can know isPrime upto 10^12 **/
+    for (LL x:primes)
+        if (N % x == 0)
+            return false;
+    return true;                    /// it takes longer time if N is a large prime!
+}
+
+
+
+/** Prime Factorization **/
+
+vector<LL> primeFactors(LL N)
+{
+    vector<LL>factors;
+    LL PF_idx = 0, PF = primes[PF_idx];
+    while (N != 1 && (PF * PF <= N))     /// stop at sqrt(N), but N can get smaller
+    {
+        while (N % PF == 0)
+        {
+            N /= PF;    /// remove this PF
+            factors.push_back(PF);
+        }
+        PF = primes[++PF_idx];                              /// only consider primes!
+    }
+    if (N != 1)
+        factors.push_back(N);     /// special case if N is actually a prime
+
+    return factors;
+}
+
+LL number_of_different_PF(LL N)
+{
+    LL PF_idx = 0, PF = primes[PF_idx], ans = 0;
+    while (N != 1 && (PF * PF <= N))
+    {
+        if (N % PF == 0)
+            ans++;                           /// count this pf only once
+        while (N % PF == 0)
+            N /= PF;
+        PF = primes[++PF_idx];
+    }
+    if (N != 1)
+        ans++;
+    return ans;
+}
+
+int main()
+{
+    optimizeIO();
+
+    bit_sieve(1e6);
+
+    while(1)
+    {
+        LL x;
+        cin>>x;
+        vector<LL>pf = primeFactors(x);
+        cout<<pf<<endl;
+        cout<<number_of_different_PF(x)<<endl;
+    }
+
+    return 0;
+}
+
+/**
+
+**/
+
+template<class T1, class T2>
+ostream &operator <<(ostream &os, pair<T1,T2>&p)
+{
+    os<<"{"<<p.first<<", "<<p.second<<"} ";
+    return os;
+}
+template <class T>
+ostream &operator <<(ostream &os, vector<T>&v)
+{
+    os<<"[ ";
+    for(int i=0; i<v.size(); i++)
+    {
+        os<<v[i]<<" " ;
+    }
+    os<<" ]";
+    return os;
+}
+
+template <class T>
+ostream &operator <<(ostream &os, set<T>&v)
+{
+    os<<"[ ";
+    for(T i:v)
+    {
+        os<<i<<" ";
+    }
+    os<<" ]";
+    return os;
+}
+
+