Question

I’m struggling with Project Euler problem 23: Non-abundant sums.

I have a script, that calculates abundant numbers:

function getSummOfDivisors( $number )
{
    $divisors = array ();
    for( $i = 1; $i < $number; $i ++ ) {
        if ( $number % $i == 0 ) {
            $divisors[] = $i;
        }
    }
    return array_sum( $divisors );
}

$limit = 28123;
//$limit  = 1000;
$matches = array();
$k = 0;

while( $k <= ( $limit/2 ) ) {
    if ( $k < getSummOfDivisors( $k ) ) {
        $matches[] = $k;
    }
    $k++;
}

echo '<pre>'; print_r( $matches );

I checked those numbers with the available on the internet already, and they are correct. I can multiply those by 2 and get the number that is the sum of two abundant numbers.

But since I need to find all numbers that cannot be written like that, I just reverse the if statement like this:

if ( $k >= getSummOfDivisors( $k ) )

This should now store all, that cannot be created as the sum of to abundant numbers, but something is not quit right here. When I sum them up I get a number that is not even close to the right answer.

I don’t want to see an answer, but I need some guidelines / tips on what am I doing wrong ( or what am I missing or miss-understanding ).

EDIT: I also tried in the reverse order, meaning, starting from top, dividing by 2 and checking if those are abundant. Still comes out wrong.

Answer 1

An error in your logic lies in the line:

“I can multiply those by 2 and get the number that is the sum of two abundant numbers”

You first determine all the abundant numbers [n1, n2, n3….] below the analytically proven limit. It is then true to state that all integers [2*n1, 2*n2,….] are the sum of two abundant numbers but n1+n2, and n2+n3 are also the sum of two abundant numbers. Therein lies your error. You have to calculate all possible integers that are the sum of any two numbers from [n1, n2, n3….] and then take the inverse to find the integers that are not.