Here's a fairly straightforward sieve. It finds solutions by repeatedly applying the basic functions to the already-found solutions, so it finds solutions for some incredibly large numbers, but doesn't get 9 (for example), likely due to the limit I put on factorial (no inputs larger than 150). Output strings are effectively unary RPN.

`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;
}
`

Update: instead of iterating a given number of times, iterates until no new solutions are found.

**Caution:** Contents may have been coded under pressure.
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