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- #!/usr/bin/perl
- # Daniel "Trizen" Șuteu
- # Date: 29 March 2021
- # https://github.com/trizen
- # Generate all the squarefree k-almost primes <= n, using a given list of prime factors.
- use 5.020;
- use ntheory qw(:all);
- use experimental qw(signatures);
- sub squarefree_almost_primes ($n, $k, $factors) {
- 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 - 1) {
- my $q = $factors->[$j];
- last if ($q > $L);
- __SUB__->(mulint($m, $q), $k - 1, $j + 1);
- }
- }->(1, $k);
- sort { $a <=> $b } @list;
- }
- my $n = 1e6; # limit
- my @factors = @{primes(17)}; # prime list
- foreach my $k (0 .. scalar(@factors)) {
- my @divisors = squarefree_almost_primes($n, $k, \@factors);
- printf("%2d-squarefree almost primes <= %s: [%s]\n", $k, $n, join(', ', @divisors));
- }
- __END__
- 0-squarefree almost primes <= 1000000: [1]
- 1-squarefree almost primes <= 1000000: [2, 3, 5, 7, 11, 13, 17]
- 2-squarefree almost primes <= 1000000: [6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 39, 51, 55, 65, 77, 85, 91, 119, 143, 187, 221]
- 3-squarefree almost primes <= 1000000: [30, 42, 66, 70, 78, 102, 105, 110, 130, 154, 165, 170, 182, 195, 231, 238, 255, 273, 286, 357, 374, 385, 429, 442, 455, 561, 595, 663, 715, 935, 1001, 1105, 1309, 1547, 2431]
- 4-squarefree almost primes <= 1000000: [210, 330, 390, 462, 510, 546, 714, 770, 858, 910, 1122, 1155, 1190, 1326, 1365, 1430, 1785, 1870, 2002, 2145, 2210, 2618, 2805, 3003, 3094, 3315, 3927, 4641, 4862, 5005, 6545, 7293, 7735, 12155, 17017]
- 5-squarefree almost primes <= 1000000: [2310, 2730, 3570, 4290, 5610, 6006, 6630, 7854, 9282, 10010, 13090, 14586, 15015, 15470, 19635, 23205, 24310, 34034, 36465, 51051, 85085]
- 6-squarefree almost primes <= 1000000: [30030, 39270, 46410, 72930, 102102, 170170, 255255]
- 7-squarefree almost primes <= 1000000: [510510]
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