The code below finds 1..10 in 2 seconds on this machine, having calculated nothing greater than 201!; 1..37 takes 8s, and calculates 366!. Beyond that we start to find some more difficult, but at 37 mins and 24MB it has so far found all but 8 numbers (59, 64, 72, 86, 87, 92, 97, 99) out of 1..99.

**Update**: still missing (92, 97, 99) after 274 mins (process size 67MB).

The basic ideas are: a) keep a sorted list (`@try`) of the numbers we've reached but not yet calculated a factorial for; b) at each iteration, take the smallest untried number and find it's factorial, and repeatedly take the square root of the result until we get to a number we've seen before; c) use a binary chop to insert new pending numbers.

`#!/usr/bin/perl
use strict;
use bigint; # optionally with eg C< lib => 'Pari' >
my $max = shift || 10;
my @try;
# deal with '1' explicitly
my %seen = (1 => 's');
my $waiting = $max - 1;
print "1 => s\n";
insert(3, '');
while (@try) {
my $n = shift @try;
my $s = 'f' . $seen{$n};
$n->bfac; # $n = ($n)!
$s = "f$s";
while (!defined $seen{$n}) {
insert($n, $s);
$n = sqrt($n);
$s = "s$s";
}
}
sub insert {
my($n, $s) = @_;
if ($n <= $max) {
print "$n => $s\n";
exit 0 unless --$waiting;
}
$seen{$n} = $s;
my($min, $max) = (0, scalar @try);
while ($min + 1 < $max) {
my $new = ($min + $max) >> 1;
if ($try[$new] > $n) {
$max = $new;
} else {
$min = $new;
}
}
++$min if $min < @try && $try[$min] < $n;
splice @try, $min, 0, $n;
}
`

In the output, eg "5 => ssff" means that 5 = int(sqrt(int(sqrt(fact(fact(3)))))).

Hugo

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