PHP Recursion Script is taking took long to execute HackerRank powerSum problem - recursion

Question

Time limit exceeded
Your code did not execute within the time limits. Please optimize your code.
<?php
/*
* Complete the 'powerSum' function below.
*
* The function is expected to return an INTEGER.
* The function accepts following parameters:
* 1. INTEGER X
* 2. INTEGER N
*/
function powerSum($X, $N) {
// Write your code here
if (1 <= $X && $X <= 1000 && 2 <= $N && $N <= 10) {
$num = 1;
$list = [];
while ( true ) {
$pow = pow($num, $N);
if ($pow > $X) {break;}
$list[] = $pow;
$num ++;
}
$limit = $X;
$array = $list;
// the algorithm is usable if the number of elements is less than 20 (because set_time_limit)
$num = count($array);
//The total number of possible combinations
$total = pow($N, $num);
$out = array();
// loop through each possible combination
for ($i = 0; $i < $total; $i++) {
$comb = array();
// for each combination check if each bit is set
for ($j = 0; $j < $num; $j++) {
// is bit $j set in $i?
if (pow($N, $j) & $i){
$comb[] = $array[$j];
}
}
if (array_sum($comb) == $limit)
{
$out[] = $comb;
}
}
array_multisort(array_map('count', $out), SORT_ASC, $out);
$out = array_unique($out, SORT_REGULAR);
return count($out);
}
}
The above function is working and passing a lot of test cases but fails only due to timeout reasons.
I hope someone will help me fix this problem as soon as possible.
I ran the code on HackerRank, it ran well but failed on test case due to time.
The link to this challenge on hackerRank is:https://www.hackerrank.com/challenges/the-power-sum/problem?isFullScreen=true