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Definition

An Erdős–Nicolas number is a positive integer which is not perfect but is equal to the sum of its first **k** divisors (arranged in ascending order and including one) for some value of **k** greater than one.

Examples

24 is an Erdős–Nicolas number because the sum of its first 6 divisors (1, 2, 3, 4, 6 and 8) is equal to 24 and it is not perfect because 12 is also a divisor.

6 is not an Erdős–Nicolas number because it is perfect (1 + 2 + 3 = 6).

48 is not an Erdős–Nicolas number because its divisors are: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. The first seven of these add up to 36, but the first eight add up to 52 which is more than 48.

Task

Find and show here the first 8 Erdős–Nicolas numbers and the number of divisors needed (i.e. the value of 'k') to satisfy the definition.

Stretch

Do the same for any further Erdős–Nicolas numbers which you have the patience for.

Note

As all known Erdős–Nicolas numbers are even you may assume this to be generally true in order to quicken up the search. However, it is not obvious (to me at least) why this should necessarily be the case.

Reference

* OEIS:A194472 - Erdős–Nicolas numbers
assistant
```
use Prime::Factor;

sub is-Erdős-Nicolas ($n) {
    my @divisors = $n.&proper-divisors: :s;
    ((@divisors.sum > $n) && (my $key = ([\+] @divisors).first: $n, :k)) ?? 1 + $key !! False
}

my $count;

(1..*).hyper(:2000batch).map( * × 2 ).map: {
    if my $key = .&is-Erdős-Nicolas {
        printf "%8d == sum of its first %3d divisors\n", $_, $key;
        exit if ++$count >= 8;
    }
}

```

Output:

```
      24 == sum of its first   6 divisors
    2016 == sum of its first  31 divisors
    8190 == sum of its first  43 divisors
   42336 == sum of its first  66 divisors
   45864 == sum of its first  66 divisors
  392448 == sum of its first  68 divisors
  714240 == sum of its first 113 divisors
 1571328 == sum of its first 115 divisors
```