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optimal algorithm for finding unique divisors

I have asked this question before, but that account is not accessible now, so I am asking it again showing my effort this time.

Given a list(array) of numbers and a number N, find the all the divisors of N which doesn’t divide any number belonging to the list. I have solved it with a brute force and a little efficient approach(but not the best one). So, I am looking for an approach which could be the best in solving this kind of problem. Everything is in terms of integer(no floating points).

My approach to this is that I first find all the divisors of the number N(without any overhead).Then, I sort the list and the divisors in reverse order(separately). Now, for each divisor D, I check if it divides any number in the list(starting from the highest element upto an element which is >= the divisor D). If it divides, then all divisors of D must also divide. Then I remove those elements from the list of divisors which are also the divisors of D(can be thought of as removing the intersection). So, ultimately the left array of divisors is the required array(according to my approach). If someone can point any fault or any lack of efficiency in my approach, it is appreciated. The max value which can be present in the list is 10^18.

I have implemented it in PHP. I am providing my code below. Please ignore the comments.

while($div=each($divisors))
{
$i=0;
$divisor=$div['key'];
//echo "divisor is $divisor\n";
while((int)$unfriendly[$i]>=$divisor)
{//echo "aya\n";
    if(!((int)bcmod($unfriendly[$i],$divisor)))
    {//echo "ayeea\n";
        $divisors_of_divisor=divisors_of_a_number($divisor);
        //print_r($divisors_of_divisor);
        //print_r($divisors);
        foreach($divisors_of_divisor as $d)
        unset($divisors[$d]);
        //print_r($divisors);
        break;
    }
    ++$i;
}
 }
echo sizeof($divisors);
function divisors_of_a_number($n)//returns all the divisors of a number in an unsorted array
{
$i=1;
$s=sqrt($n);
while($i<=$s)
{
if(!($n%$i))
{
    $a[]=$i;
    if($i!=$s)
    $a[]=$n/$i;
}
++$i;
}
return $a;
}
function divisors_of_a_number_as_keys_of_array($n)//returns all the divisors of a number in an unsorted array as keys
{
$i=1;
$s=sqrt($n);
while($i<=$s)
{
if(!($n%$i))
{
    $a[$i]=1;
    //if($i!=$s)
    $a[$n/$i]=1;
}
++$i;
}
return $a;
}

Answer 1

You can use this PHP implementation of the sieve of Eratosthenes.

And also this.

And this.

Take a look to this question.