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# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;

class Program
{
    const int M = 1000000007;
    const double eps = 1e-9;
    static int[] dd = { 0, 1, 0, -1, 0 };
    static List<long> tri;
    static void Main()
    {
        var sw = new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false };
        var sc = new Scan();
        int n = sc.Int;
        var s = new long[n];
        int lim = 200000;
        var pr = new bool[lim];
        for (int i = 2; i < lim; i++)
            pr[i] = true;
        for (int i = 2; i * i < lim; i++)
        {
            if (pr[i])
            {
                for (int j = i * i; j < lim; j += i)
                    pr[j] = false;
            }
        }
        tri = new List<long>();
        for (int i = 2; i < lim; i++)
        {
            if (pr[i])
                tri.Add(i * (long)i * i);
        }
        var dic = new SortedDictionary<long, int>();
        for (int i = 0; i < n; i++)
        {
            s[i] = conv(sc.Long);
            if (dic.ContainsKey(s[i]))
                ++dic[s[i]];
            else
                dic.Add(s[i], 1);
        }
        var key = new long[dic.Count]; // num
        var val = new int[dic.Count]; // cnt
        int cnt = 0;
        foreach (var item in dic)
        {
            key[cnt] = item.Key;
            val[cnt] = item.Value;
            ++cnt;
        }
        Array.Sort(val, key);
        var ng = new SortedSet<long>();
        int ans = 0;
        for (int i = cnt - 1; i >= 0; i--)
        {
            if (key[i] == 1)
            {
                ++ans;
                continue;
            }
            if (ng.Contains(key[i])) continue;

            ng.Add(conv(key[i] * key[i]));
            ans += val[i];
        }
        sw.WriteLine(ans);
        sw.Flush();
    }
    static long conv(long a)
    {
        long b = a;
        for (int i = 0; i < tri.Count; i++)
        {
            if (b < tri[i]) return b;
            if (b % tri[i] == 0)
            {
                b /= tri[i];
                --i;
            }
        }
        return b;
    }

    static void swap<T>(ref T a, ref T b) { var t = a; a = b; b = t; }
    static void swap<T>(IList<T> a, int i, int j) { var t = a[i]; a[i] = a[j]; a[j] = a[i]; }
    static T Max<T>(params T[] a) { return a.Max(); }
    static T Min<T>(params T[] a) { return a.Min(); }

    static T[] copy<T>(IList<T> a)
    {
        var ret = new T[a.Count];
        for (int i = 0; i < a.Count; i++) ret[i] = a[i];
        return ret;
    }
}
class Scan
{
    public int Int { get { return int.Parse(Str); } }
    public long Long { get { return long.Parse(Str); } }
    public double Double { get { return double.Parse(Str); } }
    public string Str { get { return Console.ReadLine().Trim(); } }
    public int[] IntArr { get { return StrArr.Select(int.Parse).ToArray(); } }
    public int[] IntArrWithSep(char sep) { return Str.Split(sep).Select(int.Parse).ToArray(); }
    public long[] LongArr { get { return StrArr.Select(long.Parse).ToArray(); } }
    public double[] DoubleArr { get { return StrArr.Select(double.Parse).ToArray(); } }
    public string[] StrArr { get { return Str.Split(); } }
    public void Multi(out int a, out int b) { var arr = IntArr; a = arr[0]; b = arr[1]; }
    public void Multi(out int a, out int b, out int c) { var arr = IntArr; a = arr[0]; b = arr[1]; c = arr[2]; }
    public void Multi(out int a, out string b) { var arr = StrArr; a = int.Parse(arr[0]); b = arr[1]; }
    public void Multi(out string a, out int b) { var arr = StrArr; a = arr[0]; b = int.Parse(arr[1]); }
    public void Multi(out int a, out char b) { var arr = StrArr; a = int.Parse(arr[0]); b = arr[1][0]; }
    public void Multi(out char a, out int b) { var arr = StrArr; a = arr[0][0]; b = int.Parse(arr[1]); }
    public void Multi(out long a, out long b) { var arr = LongArr; a = arr[0]; b = arr[1]; }
    public void Multi(out long a, out int b) { var arr = LongArr; a = arr[0]; b = (int)arr[1]; }
    public void Multi(out int a, out long b) { var arr = LongArr; a = (int)arr[0]; b = arr[1]; }
    public void Multi(out string a, out string b) { var arr = StrArr; a = arr[0]; b = arr[1]; }
}
class mymath
{
    static int Mod = 1000000007;
    public void setMod(int m) { Mod = m; }
    public bool isprime(long a)
    {
        if (a < 2) return false;
        for (long i = 2; i * i <= a; i++) if (a % i == 0) return false;
        return true;
    }
    public long[][] powmat(long[][] A, long n)
    {
        var E = new long[A.Length][];
        for (int i = 0; i < A.Length; i++)
        {
            E[i] = new long[A.Length];
            E[i][i] = 1;
        }
        if (n == 0) return E;
        var t = powmat(A, n / 2);
        if ((n & 1) == 0) return mulmat(t, t);
        return mulmat(mulmat(t, t), A);
    }
    public long[] mulmat(long[][] A, long[] x)
    {
        var ans = new long[A.Length];
        for (int i = 0; i < A.Length; i++) for (int j = 0; j < x.Length; j++) ans[i] = (ans[i] + x[j] * A[i][j]) % Mod;
        return ans;
    }
    public long[][] mulmat(long[][] A, long[][] B)
    {
        var ans = new long[A.Length][];
        for (int i = 0; i < A.Length; i++)
        {
            ans[i] = new long[B[0].Length];
            for (int j = 0; j < B[0].Length; j++) for (int k = 0; k < B.Length; k++) ans[i][j] = (ans[i][j] + A[i][k] * B[k][j]) % Mod;
        }
        return ans;
    }
    public long powmod(long a, long b)
    {
        if (a >= Mod) return powmod(a % Mod, b);
        if (a == 0) return 0;
        if (b == 0) return 1;
        var t = powmod(a, b / 2);
        if ((b & 1) == 0) return t * t % Mod;
        return t * t % Mod * a % Mod;
    }
    public long gcd(long a, long b)
    {
        while (b > 0) { var t = a % b; a = b; b = t; }
        return a;
    }
    public long lcm(long a, long b) { return a * (b / gcd(a, b)); }
    public long Comb(int n, int r)
    {
        if (n < 0 || r < 0 || r > n) return 0;
        if (n - r < r) r = n - r;
        if (r == 0) return 1;
        if (r == 1) return n;
        var numerator = new int[r];
        var denominator = new int[r];
        for (int k = 0; k < r; k++)
        {
            numerator[k] = n - r + k + 1;
            denominator[k] = k + 1;
        }
        for (int p = 2; p <= r; p++)
        {
            int pivot = denominator[p - 1];
            if (pivot > 1)
            {
                int offset = (n - r) % p;
                for (int k = p - 1; k < r; k += p)
                {
                    numerator[k - offset] /= pivot;
                    denominator[k] /= pivot;
                }
            }
        }
        long result = 1;
        for (int k = 0; k < r; k++) if (numerator[k] > 1) result = result * numerator[k] % Mod;
        return result;
    }
}