use strict;
use warnings;
use bigint;
# Each element of @formulas is an an op-string:
# a string of Fs and Ss, indicating factorial
# and square root, in order applied to get the
# index
my %formulas = (3 => '3');
my $found_new = 1;
while ($found_new) {
    $found_new = 0;
    # Apply factorial to all formulas
    while (my ($v, $ops) = each %formulas) {
        if ($v < 150) {
          $v = fact($v);
          $ops .= 'F';
          exists $formulas{$v} or $found_new = $formulas{$v} = $ops;
        }
    }
    # Apply sqrt to all formulas
    while (my ($v, $ops) = each %formulas) {
        $v = int(sqrt($v));
        $ops .= 'S';
        exists $formulas{$v} or $found_new = $formulas{$v} = $ops;
    }
}

for (sort { $a <=> $b } keys %formulas) {
    print "$_: $formulas{$_}\n";
}

sub fact {
    my $f = 1;
    for (2..$_[0]) { $f *= $_ }
    $f;
}