Step by Step solution about How to resolve the algorithm Proper divisors step by step in the PHP programming language

Explanation:

This PHP code defines a function called ProperDivisors that generates all the proper divisors of a given number n. Proper divisors are the positive divisors of n that are less than n itself.

Function Implementation:

• The function uses a generator to yield (output) the divisors one by one.
• It starts by yielding 1, which is a proper divisor of every number.
• It then iterates over the range [2, sqrt(n)].
• For each i in this range, it checks if n % i == 0. If i divides n evenly, it yields i and also adds n / i to a list of large divisors if i is not the square root of n.
• After the loop, it iterates over the large divisors in reverse order and yields them.

Assertions and Example Output:

• The code includes an assertion that verifies the correctness of the function for n = 100.
• It also prints the proper divisors of numbers from 1 to 10.

Counting Divisors:

• The code initializes an array called $divisorsCount.
• It iterates over the range [1, 20000) and counts the number of proper divisors for each number i.
• It stores the count along with the numbers having that count in the $divisorsCount array.
• Finally, it sorts $divisorsCount by the number of divisors and prints the numbers with the most divisors and their count.

<?php
function ProperDivisors($n) {
  yield 1;
  $large_divisors = [];
  for ($i = 2; $i <= sqrt($n); $i++) {
    if ($n % $i == 0) {
      yield $i;
      if ($i*$i != $n) {
        $large_divisors[] = $n / $i;
      }
    }
  }
  foreach (array_reverse($large_divisors) as $i) {
    yield $i;
  }
}

assert([1, 2, 4, 5, 10, 20, 25, 50] ==
        iterator_to_array(ProperDivisors(100)));

foreach (range(1, 10) as $n) {
  echo "$n =>";
  foreach (ProperDivisors($n) as $divisor) {
    echo " $divisor";
  }
  echo "\n";
}

$divisorsCount = [];
for ($i = 1; $i < 20000; $i++) {
  $divisorsCount[sizeof(iterator_to_array(ProperDivisors($i)))][] = $i;
}
ksort($divisorsCount);

echo "Numbers with most divisors: ", implode(", ", end($divisorsCount)), ".\n";
echo "They have ", key($divisorsCount), " divisors.\n";