Problem:
Score : 1100 points

Problem Statement

Snuke got positive integers s_1,...,s_N from his mother, as a birthday present. There may be duplicate elements.

He will circle some of these N integers. Since he dislikes cubic numbers, he wants to ensure that if both s_i and s_j (i ≠ j) are circled, the product s_is_j is not cubic. For example, when s_1=1,s_2=1,s_3=2,s_4=4, it is not possible to circle both s_1 and s_2 at the same time. It is not possible to circle both s_3 and s_4 at the same time, either.

Find the maximum number of integers that Snuke can circle.

Constraints

1 ≦ N ≦ 10^5

1 ≦ s_i ≦ 10^{10}

All input values are integers.

Input

The input is given from Standard Input in the following format:

N
s_1
:
s_N

Output

Print the maximum number of integers that Snuke can circle.

Sample Input 1

8
1
2
3
4
5
6
7
8

Sample Output 1

6

Snuke can circle 1,2,3,5,6,7.

Sample Input 2

6
2
4
8
16
32
64

Sample Output 2

3

Sample Input 3

10
1
10
100
1000000007
10000000000
1000000009
999999999
999
999
999

Sample Output 3

9

Solution in C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <iostream>
#include <vector>
#include <cstdio>
#include <sstream>
#include <map>
#include <string>
#include <algorithm>
#include <queue>
#include <cmath>
#include <functional>
#include <set>
#include <ctime>
#include <random>
#include <chrono>
#include <cassert>
using namespace std;

namespace {
  using Integer = long long; //__int128;
  template<class T> istream& operator >> (istream& is, vector<T>& vec){for(T& val: vec) is >> val; return is;}
  template<class T> istream& operator ,  (istream& is, T& val){ return is >> val;}
  template<class T> ostream& operator << (ostream& os, const vector<T>& vec){for(int i=0; i<vec.size(); i++) os << vec[i] << (i==vec.size()-1?"":" "); return os;}
  template<class T> ostream& operator ,  (ostream& os, const T& val){ return os << " " << val;}

  template<class H> void print(const H& head){ cout << head; }
  template<class H, class ... T> void print(const H& head, const T& ... tail){ cout << head << " "; print(tail...); }
  template<class ... T> void println(const T& ... values){ print(values...); cout << endl; }

  template<class H> void eprint(const H& head){ cerr << head; }
  template<class H, class ... T> void eprint(const H& head, const T& ... tail){ cerr << head << " "; eprint(tail...); }
  template<class ... T> void eprintln(const T& ... values){ eprint(values...); cerr << endl; }

  class range{
    long long start_, end_, step_;
   public:
    struct range_iterator{
      long long val, step_;
      range_iterator(long long v, long long step) : val(v), step_(step) {}
      long long operator * (){return val;}
      void operator ++ (){val += step_;}
      bool operator != (range_iterator& x){return step_ > 0 ? val < x.val : val > x.val;}
    };
    range(long long len) : start_(0), end_(len), step_(1) {}
    range(long long start, long long end) : start_(start), end_(end), step_(1) {}
    range(long long start, long long end, long long step) : start_(start), end_(end), step_(step) {}
    range_iterator begin(){ return range_iterator(start_, step_); }
    range_iterator   end(){ return range_iterator(  end_, step_); }
  };

  string operator "" _s (const char* str, size_t size){ return move(string(str)); }
  constexpr Integer my_pow(Integer x, Integer k, Integer z=1){return k==0 ? z : k==1 ? z*x : (k&1) ? my_pow(x*x,k>>1,z*x) : my_pow(x*x,k>>1,z);}
  constexpr Integer my_pow_mod(Integer x, Integer k, Integer M, Integer z=1){return k==0 ? z%M : k==1 ? z*x%M : (k&1) ? my_pow_mod(x*x%M,k>>1,M,z*x%M) : my_pow_mod(x*x%M,k>>1,M,z);}
  constexpr unsigned long long operator "" _ten (unsigned long long value){ return my_pow(10,value); }

  inline int k_bit(Integer x, int k){return (x>>k)&1;} //0-indexed

  mt19937 mt(chrono::duration_cast<chrono::nanoseconds>(chrono::steady_clock::now().time_since_epoch()).count());

  template<class T> string join(const vector<T>& v, const string& sep){
    stringstream ss; for(int i=0; i<v.size(); i++){ if(i>0) ss << sep; ss << v[i]; } return ss.str();
  }

  string operator * (string s, int k){ string ret; while(k){ if(k&1) ret += s; s += s; k >>= 1; } return ret; }

}

constexpr long long mod = 9_ten + 7;


vector<long long> Eratosthenes(long long N){
  vector<bool> v(N+1, true);
  v[0] = v[1] = false;
  long long sqN = sqrt(N);
  for(int i=2; i<=sqN; i++){
    if(v[i] == false) continue;
    for(long long j=i*i; j<=N; j+=i){
      v[j] = false;
    }
  }
  vector<long long> Prime;
  for(long long i=2; i<=N; i++){
    if(v[i]==true) Prime.push_back(i);
  }
  return Prime;
}


int main(){
  int n;
  cin >> n;
  vector<long long> v(n);
  for(int i=0; i<n; i++){
    scanf("%lld", &v[i]);
  }
  sort(v.begin(), v.end());

  auto P = Eratosthenes(100000);

  vector<long long> cube;
  for(long long i=1; i*i*i <= 10_ten; i++){
    cube.push_back(i*i*i);
  }

  vector<vector<long long>> vp(n);
  vector<vector<long long>> vk(n);

  long long ans = 0;
  bool zero = false;
  map<int, long long> u;

  for(auto i : range(n) ){
    if(v[i] == 1) zero = true;
    else if( binary_search(cube.begin(), cube.end(), v[i] ) ) zero = true;
    else u[i] = v[i];
  }

  if(zero) ans++;

  map<vector<long long>, map<vector<long long>, long long>> G;

  for(auto p : P){
    if(u.size() == 0) break;
    map<int, long long> u_;
    for(auto& x : u){
      if(x.second % p != 0){
        u_.insert( x );
        continue;
      }

      int cnt = 0;
      while(x.second % p == 0){
        x.second /= p;
        cnt++;
      }

      if(cnt%3 != 0){
        vp[x.first].push_back( p );
        vk[x.first].push_back( cnt%3 );
      }

      if(x.second != 1){
        u_.insert(x);
      }
    }

    swap(u, u_);
  }

  for(auto& x : u){
    vp[x.first].push_back( x.second );
    vk[x.first].push_back( 1 );
  }

  for(int i=0; i<n; i++){
    if(vp[i].size() == 0) continue;
    G[ vp[i] ][ vk[i] ] += 1;
  }

  for(auto& ps : G){
    set<vector<long long>> used;
    for(auto& pr : ps.second){
      if( used.count( pr.first ) ) continue;
      auto inv = pr.first;
      for(auto& w : inv){
        w = w == 1 ? 2 : 1;
      }

      if( ps.second.count( inv ) == 0 ){
        ans += pr.second;
      }else{
        ans += max(pr.second, ps.second[inv]);
        used.insert(inv);
      }
    }
  }

  println(ans);

  return 0;
}