almost_primes_from_factor_list.pl

#!/usr/bin/perl

# Daniel "Trizen" Șuteu
# Date: 29 March 2021
# https://github.com/trizen

# Generate all the possible k-almost primes <= n, using a given list of prime factors.

use 5.020;
use ntheory qw(:all);
use experimental qw(signatures);

sub almost_primes ($n, $k, $factors, $squarefree = 0) {

    my $factors_end = $#{$factors};

    if ($k == 0) {
        return (1);
    }

    if ($k == 1) {
        return @$factors;
    }

    my @list;

    sub ($m, $k, $i = 0) {

        if ($k == 1) {

            my $L = divint($n, $m);

            foreach my $j ($i .. $factors_end) {
                my $q = $factors->[$j];
                last if ($q > $L);
                push(@list, mulint($m, $q));
            }

            return;
        }

        my $L = rootint(divint($n, $m), $k);

        foreach my $j ($i .. $factors_end) {
            my $q = $factors->[$j];
            last if ($q > $L);
            __SUB__->(mulint($m, $q), $k - 1, $j + $squarefree);
        }
    }->(1, $k);

    sort { $a <=> $b } @list;
}

my $n       = 1e3;              # limit
my @factors = @{primes(11)};    # prime list

foreach my $k (0 .. scalar(@factors)) {
    my @divisors = almost_primes($n, $k, \@factors);
    printf("%2d-almost primes <= %s: [%s]\n", $k, $n, join(', ', @divisors));
}

__END__
 0-almost primes <= 1000: [1]
 1-almost primes <= 1000: [2, 3, 5, 7, 11]
 2-almost primes <= 1000: [4, 6, 9, 10, 14, 15, 21, 22, 25, 33, 35, 49, 55, 77, 121]
 3-almost primes <= 1000: [8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 63, 66, 70, 75, 98, 99, 105, 110, 125, 147, 154, 165, 175, 231, 242, 245, 275, 343, 363, 385, 539, 605, 847]
 4-almost primes <= 1000: [16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 126, 132, 135, 140, 150, 189, 196, 198, 210, 220, 225, 250, 294, 297, 308, 315, 330, 350, 375, 441, 462, 484, 490, 495, 525, 550, 625, 686, 693, 726, 735, 770, 825, 875]
 5-almost primes <= 1000: [32, 48, 72, 80, 108, 112, 120, 162, 168, 176, 180, 200, 243, 252, 264, 270, 280, 300, 378, 392, 396, 405, 420, 440, 450, 500, 567, 588, 594, 616, 630, 660, 675, 700, 750, 882, 891, 924, 945, 968, 980, 990]