This is a great idea. I always wanted to implement a sieve that can just go on finding new primes, i.e. without specifying a maximum up to which the sieving will be conducted. This would require not only to store the primes up to a point but also kind of the largest multiple of each prime up to that point. Also, I think using the modulus operator % violates the sieving philosophy. I never was able to make it work as I got confused keeping the primes and the multiples.

Using closures is ideal for that. Every time one finds a new prime, one generates a new sifter that is responsible for that prime and to store the progress to date. Here is my formulation:

use strict;
use warnings;

sub sifter {
    my $p = shift;
    my $c = $p;
    return sub { $c += $p while $c < $_[0]; return $c-$_[0]; }
}

my @sieves;
my $n = 1;

loop: while( $n++ ){
    $_->( $n ) or next loop for @sieves;
    push @sieves, sifter( $n );
    print "$n\n";
}
[download]

Clearly this is not a fast algorithm to find primes but I liked the idea...