Some Sum of Some Sum

If a number's prime factorization is $p_{1}^{q_{1}}\times p_{2}^{q_{2}}\times p_{3}^{q_{3}}\times...\times p_{k}^{q_{k}}$ , where $p_{1},p_{2},p_{3},...,p_{k}$ are primes, its sum of divisors is $(p_{1}^{0}+p_{1}^{1}+...+p_{1}^{q_{1}})(p_{2}^{0}+p_{2}^{1}+...+p_{2}^{q_{2}})...(p_{k}^{0}+p_{k}^{1}+...+p_{k}^{q_{k}})$ .

As the integers are square-free, their prime factorizations do not contain any exponents greater than 1. Therefore, their sum of divisors will be $(1+p_{1})(1+p_{2})...(1+p_{k})$ .

$288=2^{5}\times3^{2}$ . There are a total of $6$ ways to express $288$ as the product $(1+p_{1})(1+p_{2})...(1+p_{k})$ :

$288=(1+3)(1+71)$ , giving the number $3\times71=213$

$288=(1+5)(1+47)$ , giving the number $5\times47=235$

$288=(1+11)(1+23)$ , giving the number $11\times23=253$

$288=(1+2)(1+3)(1+23)$ , giving the number $2\times3\times23=138$

$288=(1+2)(1+7)(1+11)$ , giving the number $2\times7\times11=154$

$288=(1+3)(1+5)(1+11)$ , giving the number $3\times5\times11=165$

The sum of these numbers is $\boxed{1158}$ .

PHP solution, half elegant, half brute force

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<?php
$t = 0;
for($i=2;$i<=288;$i++){
    if(squarefree($i)){
        $factors = factors($i);
        $sum = array_sum($factors);
        if($sum==288){
            $t += $i;

        echo 'i: '.$i.'<br>';
        echo 'factors: '.implode(', ',$factors).'<br>';
        echo 'sum: '.$sum.'<br><br>';
        }
    }
}
echo 'answer: '.$t;


function squarefree($n){
    $i = 1;
    do {
        $i++;
        if($n%pow($i,2)==0){
            return false;
        }
    } while($n>=pow($i,2));
    return true;
}

function factors($n){
    $f = array();
    for($i=1;$i<=$n/2;$i++){
        if($n%$i==0){
            $f[] = $i;
            $f[] = $n/$i;
        }
    }
    $f = array_unique($f);
    sort($f);
    return $f;
}
/*
i: 138
factors: 1, 2, 3, 6, 23, 46, 69, 138
sum: 288

i: 154
factors: 1, 2, 7, 11, 14, 22, 77, 154
sum: 288

i: 165
factors: 1, 3, 5, 11, 15, 33, 55, 165
sum: 288

i: 213
factors: 1, 3, 71, 213
sum: 288

i: 235
factors: 1, 5, 47, 235
sum: 288

i: 253
factors: 1, 11, 23, 253
sum: 288

answer: 1158
*/
?>