code(java): Add solution for problem-2

Author: dwivediprashant

Problems/Mathematics/Day-01/sol/Prashant_dwivedi/Solution2.java

@@ -0,0 +1,104 @@
+/*
+Problem Statement:
+You are given an integer n and an integer k (k ≥ 2), along with an array of n
+positive integers.
+
+Your task is to count the number of unordered index pairs (i, j) such that:
+1 ≤ i < j ≤ n
+The product ai × aj is a perfect k-th power.
+
+---------------------------------------
+Approach:
+A number is a perfect k-th power if, in its prime factorization,
+the exponent of every prime is divisible by k.
+
+For each number:
+1. Factorize it using smallest prime factor (SPF).
+2. Reduce each prime’s exponent modulo k.
+3. Store this reduced form as a "signature".
+
+For a pair (ai, aj) to form a perfect k-th power:
+- Their signatures must complement each other such that
+  the sum of exponents (mod k) becomes 0 for every prime.
+
+---------------------------------------
+Time Complexity:
+O(n log a)
+
+Space Complexity:
+O(n)
+---------------------------------------
+*/
+
+import java.util.*;
+
+public class Solution2 {
+
+    static int MAX = 100000;
+    static int[] spf = new int[MAX + 1];
+
+    // Precompute smallest prime factor for every number
+    static void computeSPF() {
+        for (int i = 1; i <= MAX; i++)
+            spf[i] = i;
+
+        for (int i = 2; i * i <= MAX; i++) {
+            if (spf[i] == i) {
+                for (int j = i * i; j <= MAX; j += i) {
+                    if (spf[j] == j)
+                        spf[j] = i;
+                }
+            }
+        }
+    }
+
+    // Build reduced prime-exponent signature
+    static Map<Integer, Integer> buildSignature(int x, int k) {
+        Map<Integer, Integer> map = new HashMap<>();
+        while (x > 1) {
+            int p = spf[x];
+            int cnt = 0;
+            while (x % p == 0) {
+                x /= p;
+                cnt++;
+            }
+            cnt %= k;
+            if (cnt > 0)
+                map.put(p, cnt);
+        }
+        return map;
+    }
+
+    // Build complement signature
+    static Map<Integer, Integer> buildComplement(Map<Integer, Integer> sig, int k) {
+        Map<Integer, Integer> comp = new HashMap<>();
+        for (Map.Entry<Integer, Integer> e : sig.entrySet()) {
+            comp.put(e.getKey(), (k - e.getValue()) % k);
+        }
+        return comp;
+    }
+
+    public static void main(String[] args) {
+        Scanner sc = new Scanner(System.in);
+        computeSPF();
+
+        int n = sc.nextInt();
+        int k = sc.nextInt();
+
+        Map<Map<Integer, Integer>, Long> freq = new HashMap<>();
+        long ans = 0;
+
+        for (int i = 0; i < n; i++) {
+            int x = sc.nextInt();
+
+            Map<Integer, Integer> sig = buildSignature(x, k);
+            Map<Integer, Integer> comp = buildComplement(sig, k);
+
+            ans += freq.getOrDefault(comp, 0L);
+            freq.put(sig, freq.getOrDefault(sig, 0L) + 1);
+        }
+
+        System.out.println(ans);
+        sc.close();
+    }
+}