Hi Andrew,
I have rewritten your script(same as yours),
I am not sure whether my output is correct or not,
I just tried to get the output Result, What you are trying now
I hope this might help

use strict;
use warnings;
my ($i,$j,$l,$s);
my (@num,@numbers);

for $i(100..999)
    {
    undef(@num);
    @num=split(//, $i);
    foreach $j(@num){ $j=$j**3;}           
    $s=0; 

    for $l(@num) {$s = $s + $l;}
    if( $s == "$i") {push (@numbers, $i);}
    }
print @numbers;
#output is
153370371407
[download]

Thanks,
perlsen
UPDATE:
This script was written by me to produce the output for your program only.
I am not clearly go through your script.
Any way, This is waste of my precious time.
and Thanks for some fellow monks good response.