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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.InputMismatchException;

public class Main {
	static InputStream is;
	static PrintWriter out;
	static String INPUT = "";
	
	static void solve()
	{
		int n = ni();
		int[] primes = sieveEratosthenes(3000);
		long[] a = new long[n];
		for(int i = 0;i < n;i++){
			a[i] = nl();
			for(int p : primes){
				long q = (long)p*p*p;
				if(q > a[i])break;
				while(a[i] % q == 0){
					a[i] /= q;
				}
			}
		}
		Arrays.sort(a);
		long[][] cs = new long[n][];
		long[] cv = new long[n];
		int q = 0;
		for(int i = 0;i < n;){
			int j = i;
			while(j < n && a[j] == a[i])j++;
			cv[q] = a[i];
			cs[q++] = new long[]{a[i], j-i};
			i = j;
		}
		long ret = 0;
		for(int i = 0;i < q;i++){
			if(cs[i][0] == 1){
				ret += 1;
			}else{
				ret += cs[i][1];
				long[][] f = factor(cs[i][0], primes);
				if(f == null){
					continue;
				}
				long fc = 1;
				long qc = 1;
				for(long[] u : f){
					fc *= u[0];
					for(int j = 0;j < 3-u[1];j++){
						qc *= u[0];
					}
				}
				if((double)fc*fc*fc/cs[i][0] > 1e11){
					continue;
				}
				int ind = Arrays.binarySearch(cv, 0, q, qc);
				if(ind >= 0 && ind > i){
					ret -= Math.min(cs[ind][1], cs[i][1]);
				}
			}
		}
		out.println(ret);
	}
	
	public static long[][] factor(long n, int[] primes)
	{
		long[][] ret = new long[15][2];
		int rp = 0;
		for(int p : primes){
			if((long)p * p > n)break;
			int i;
			for(i = 0;n % p == 0;n /= p, i++);
			if(i > 0){
				ret[rp][0] = p;
				ret[rp][1] = i;
				rp++;
			}
		}
		if(n != 1){
			if(n > 3000)return null;
			ret[rp][0] = n;
			ret[rp][1] = 1;
			rp++;
		}
		return Arrays.copyOf(ret, rp);
	}

	
	public static int[] sieveEratosthenes(int n) {
		if (n <= 32) {
			int[] primes = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 };
			for (int i = 0; i < primes.length; i++) {
				if (n < primes[i]) {
					return Arrays.copyOf(primes, i);
				}
			}
			return primes;
		}

		int u = n + 32;
		double lu = Math.log(u);
		int[] ret = new int[(int) (u / lu + u / lu / lu * 1.5)];
		ret[0] = 2;
		int pos = 1;

		int[] isnp = new int[(n + 1) / 32 / 2 + 1];
		int sup = (n + 1) / 32 / 2 + 1;

		int[] tprimes = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 };
		for (int tp : tprimes) {
			ret[pos++] = tp;
			int[] ptn = new int[tp];
			for (int i = (tp - 3) / 2; i < tp << 5; i += tp)
				ptn[i >> 5] |= 1 << (i & 31);
			for (int j = 0; j < sup; j += tp) {
				for (int i = 0; i < tp && i + j < sup; i++) {
					isnp[j + i] |= ptn[i];
				}
			}
		}

		// 3,5,7
		// 2x+3=n
		int[] magic = { 0, 1, 23, 2, 29, 24, 19, 3, 30, 27, 25, 11, 20, 8, 4, 13, 31, 22, 28, 18, 26, 10, 7, 12, 21, 17,
				9, 6, 16, 5, 15, 14 };
		int h = n / 2;
		for (int i = 0; i < sup; i++) {
			for (int j = ~isnp[i]; j != 0; j &= j - 1) {
				int pp = i << 5 | magic[(j & -j) * 0x076be629 >>> 27];
				int p = 2 * pp + 3;
				if (p > n)
					break;
				ret[pos++] = p;
				if ((long) p * p > n)
					continue;
				for (int q = (p * p - 3) / 2; q <= h; q += p)
					isnp[q >> 5] |= 1 << q;
			}
		}

		return Arrays.copyOf(ret, pos);
	}

	
	public static int lowerBound(long[] a, long v)
	{
		int low = -1, high = a.length;
		while(high-low > 1){
			int h = high+low>>>1;
			if(a[h] >= v){
				high = h;
			}else{
				low = h;
			}
		}
		return high;
	}

	
	public static void main(String[] args) throws Exception
	{
//		int n = 100000, m = 99999;
//		Random gen = new Random();
//		StringBuilder sb = new StringBuilder();
//		sb.append(n + " ");
//		for (int i = 0; i < n; i++) {
//			sb.append(Math.abs(gen.nextLong() % 10000000L)+1 + " ");
//		}
//		INPUT = sb.toString();

		
		long S = System.currentTimeMillis();
		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);
		
		solve();
		out.flush();
		long G = System.currentTimeMillis();
		tr(G-S+"ms");
	}
	
	private static boolean eof()
	{
		if(lenbuf == -1)return true;
		int lptr = ptrbuf;
		while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
		
		try {
			is.mark(1000);
			while(true){
				int b = is.read();
				if(b == -1){
					is.reset();
					return true;
				}else if(!isSpaceChar(b)){
					is.reset();
					return false;
				}
			}
		} catch (IOException e) {
			return true;
		}
	}
	
	private static byte[] inbuf = new byte[1024];
	static int lenbuf = 0, ptrbuf = 0;
	
	private static int readByte()
	{
		if(lenbuf == -1)throw new InputMismatchException();
		if(ptrbuf >= lenbuf){
			ptrbuf = 0;
			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
			if(lenbuf <= 0)return -1;
		}
		return inbuf[ptrbuf++];
	}
	
	private static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
//	private static boolean isSpaceChar(int c) { return !(c >= 32 && c <= 126); }
	private static int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
	
	private static double nd() { return Double.parseDouble(ns()); }
	private static char nc() { return (char)skip(); }
	
	private static String ns()
	{
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while(!(isSpaceChar(b))){
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	
	private static char[] ns(int n)
	{
		char[] buf = new char[n];
		int b = skip(), p = 0;
		while(p < n && !(isSpaceChar(b))){
			buf[p++] = (char)b;
			b = readByte();
		}
		return n == p ? buf : Arrays.copyOf(buf, p);
	}
	
	private static char[][] nm(int n, int m)
	{
		char[][] map = new char[n][];
		for(int i = 0;i < n;i++)map[i] = ns(m);
		return map;
	}
	
	private static int[] na(int n)
	{
		int[] a = new int[n];
		for(int i = 0;i < n;i++)a[i] = ni();
		return a;
	}
	
	private static int ni()
	{
		int num = 0, b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static long nl()
	{
		long num = 0;
		int b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
}