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.Linq;
using System.Linq.Expressions;
using System.Collections.Generic;
using Debug = System.Diagnostics.Debug;
using StringBuilder = System.Text.StringBuilder;
using System.Numerics;
using Number = System.Int64;
namespace Program
{
    public class Solver
    {
        public void Solve()
        {
            var n = sc.Integer();
            var primes = MathEx.Sieve(200000);
            var a = sc.Long(n);
            var map = new SortedMap<long, int>();
            var pp = MathEx.SieveList(2200);
            foreach (long p in pp)
            {
                var q = p * p * p;
                for (int i = 0; i < n; i++)
                {
                    while (a[i] % q == 0) a[i] /= q;
                }
            }
            Debug.WriteLine(a.AsJoinedString());
            var used = new HashSet<long>();
            foreach (var x in a) map[x]++;

            var ans = 0;
            const long otaku = 10000000001;
            foreach (var x in map)
            {

                var y = x.Key;
                if (y == 1) { ans++; continue; }
                if (used.Contains(y)) continue;
                Debug.WriteLine(x);
                var inv = 1L;
                foreach (long p in pp)
                {
                    var q = p * p;
                    if (y % q == 0) { inv = Math.Min(otaku, inv * p); y /= q; }
                    else if (y % p == 0)
                    {
                        inv = Math.Min(otaku, inv * p);
                        inv = Math.Min(otaku, inv * p);
                        y /= p;
                    }
                }

                var u = x.Value;
                if (y != 1)
                {
                    if (y < 200000 && primes[y])
                    {
                        y = 1;
                        inv = Math.Min(otaku, inv * y);
                        inv = Math.Min(otaku, inv * y);
                    }
                    else
                    {
                        inv = otaku;
                    }
                }
                int v;
                if (map.TryGetValue(inv, out v))
                {
                    ans += Math.Max(u, v);
                    used.Add(inv);
                }
                else ans += u;

                used.Add(x.Key);

            }
            IO.Printer.Out.WriteLine(ans);
        }
        public IO.StreamScanner sc = new IO.StreamScanner(Console.OpenStandardInput());
        static T[] Enumerate<T>(int n, Func<int, T> f) { var a = new T[n]; for (int i = 0; i < n; ++i) a[i] = f(i); return a; }
        static public void Swap<T>(ref T a, ref T b) { var tmp = a; a = b; b = tmp; }


    }
}
#region main
static class Ex
{
    static public string AsString(this IEnumerable<char> ie) { return new string(System.Linq.Enumerable.ToArray(ie)); }
    static public string AsJoinedString<T>(this IEnumerable<T> ie, string st = " ") { return string.Join(st, ie); }
    static public void Main()
    {
        var solver = new Program.Solver();
        solver.Solve();
        Program.IO.Printer.Out.Flush();
    }
}
#endregion
#region Ex
namespace Program.IO
{
    using System.IO;
    using System.Text;
    using System.Globalization;
    public class Printer: StreamWriter
    {
        static Printer() { Out = new Printer(Console.OpenStandardOutput()) { AutoFlush = false }; }
        public static Printer Out { get; set; }
        public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
        public Printer(System.IO.Stream stream) : base(stream, new UTF8Encoding(false, true)) { }
        public Printer(System.IO.Stream stream, Encoding encoding) : base(stream, encoding) { }
        public void Write<T>(string format, T[] source) { base.Write(format, source.OfType<object>().ToArray()); }
        public void WriteLine<T>(string format, T[] source) { base.WriteLine(format, source.OfType<object>().ToArray()); }
    }
    public class StreamScanner
    {
        public StreamScanner(Stream stream) { str = stream; }
        public readonly Stream str;
        private readonly byte[] buf = new byte[1024];
        private int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        private byte read()
        {
            if (isEof) return 0;
            if (ptr >= len) { ptr = 0; if ((len = str.Read(buf, 0, 1024)) <= 0) { isEof = true; return 0; } }
            return buf[ptr++];
        }
        public char Char() { byte b = 0; do b = read(); while ((b < 33 || 126 < b) && !isEof); return (char)b; }

        public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        public string ScanLine()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b != '\n'; b = (char)read())
                if (b == 0) break;
                else if (b != '\r') sb.Append(b);
            return sb.ToString();
        }
        public long Long()
        {
            if (isEof) return long.MinValue;
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != 0 && b != '-' && (b < '0' || '9' < b));
            if (b == 0) return long.MinValue;
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        public int Integer() { return (isEof) ? int.MinValue : (int)Long(); }
        public double Double() { var s = Scan(); return s != "" ? double.Parse(s, CultureInfo.InvariantCulture) : double.NaN; }
        private T[] enumerate<T>(int n, Func<T> f)
        {
            var a = new T[n];
            for (int i = 0; i < n; ++i) a[i] = f();
            return a;
        }

        public char[] Char(int n) { return enumerate(n, Char); }
        public string[] Scan(int n) { return enumerate(n, Scan); }
        public double[] Double(int n) { return enumerate(n, Double); }
        public int[] Integer(int n) { return enumerate(n, Integer); }
        public long[] Long(int n) { return enumerate(n, Long); }
    }
}
#endregion

#region sieve
static public partial class MathEx
{
    static public bool[] Sieve(int p)
    {
        var isPrime = new bool[p + 1];
        for (int i = 2; i <= p; i++) isPrime[i] = true;
        for (int i = 2; i * i <= p; i++)
            if (!isPrime[i]) continue;
            else
                for (int j = i * i; j <= p; j += i)
                    isPrime[j] = false;
        return isPrime;
    }
}
#endregion

#region sieve list
static public partial class MathEx
{
    static public List<int> SieveList(int p)
    {
        var isPrime = new bool[p + 1];
        for (int i = 2; i <= p; i++)
            isPrime[i] = true;
        for (int i = 2; i * i <= p; i++)
        {
            if (!isPrime[i])
                continue;
            for (int j = i * i; j <= p; j += i)
                isPrime[j] = false;
        }
        var ret = new List<int>();
        for (int i = 0; i <= p; i++) if (isPrime[i]) ret.Add(i);
        return ret;
    }
}
#endregion

#region SortedMap
class SortedMap<K, V>: SortedDictionary<K, V>
{
    new public V this[K i]
    {
        get
        {
            V v;
            return TryGetValue(i, out v) ? v : base[i] = default(V);
        }
        set { base[i] = value; }
    }
}
#endregion