I'm just curious ... why are you using a hash? Hashes are great for string-based indexes or for sparse arrays. But here we have a contiguous array from 1 to 20,000. An array would not only be smaller, but faster. Lots faster. Even with your hash, we could get a bit more out of looping less. In your print, I'm not sure why you grep from the map. You could easily combine them as:

print join(', ', map { $lights{ $_ } ? $_ : () } 
                     ( 1..20_000)
      )."\n";
[download]

Even that is sub-optimal. After all, you want the original number - so you could just grep it out. I've put it as duckyd2 in my benchmark. And, for good measure, I've also added tanktalus as just the rewrite to use arrays.

#!/usr/bin/perl
use strict;
use warnings;

use Benchmark qw(cmpthese);

my $duckyd_answer;

sub duckyd {
    my %lights = map { $_ => 1 } ( 1..20_000 );

    foreach my $flipper ( 2..20_000) {
        for( my $i = $flipper; $i <= 20_000; $i += $flipper ){
            $lights{ $i } = !$lights{ $i };
        }
    }

    $duckyd_answer = join(', ', grep { defined $_ }
                          map { $lights{ $_ } ? $_ : undef }
                          ( 1..20_000)
                         );
}

my $duckyd2_answer;

sub duckyd2 {
    my %lights = map { $_ => 1 } ( 1..20_000 );

    foreach my $flipper ( 2..20_000) {
        for( my $i = $flipper; $i <= 20_000; $i += $flipper ){
            $lights{ $i } = !$lights{ $i };
        }
    }

    $duckyd2_answer = join(', ', grep { $lights{ $_ } }
                           ( 1..20_000)
                          );
}

my $tanktalus_answer;

sub tanktalus {
    my @lights = (1) x 20_000;

    foreach my $flipper ( 2..20_000) {
        for( my $i = $flipper; $i <= 20_000; $i += $flipper ){
            $lights[$i-1] = !$lights[$i-1];
        }
    }

    $tanktalus_answer = join(', ', grep { $lights[$_-1] } 1..20_000);
}

cmpthese(-1,
         {
             duckyd    => \&duckyd,
             duckyd2   => \&duckyd2,
             tanktalus => \&tanktalus,
         },
        );

print "duckyd   answer: $duckyd_answer\n";
print "duckyd2  answer: $duckyd2_answer\n";
print "tankalus answer: $tanktalus_answer\n";
[download]

And the results on this machine:

            Rate    duckyd   duckyd2 tanktalus
duckyd    47.3/s        --       -1%      -44%
duckyd2   47.7/s        1%        --      -44%
tanktalus 84.6/s       79%       77%        --
[download]

(I've removed the answers as they're all the same.) The 1% speed benefit of duckyd2 over duckyd is purely the removal of the map in the output string - mostly ignorable, I grant, as it's still O(n) vs the O(n^2) algorithm right before it. As you can see, arrays are significantly faster than hashes for this. Still O(n^2), but the constant is reduced ;-)