Pierpont primes
Finesse version
This finesse version never produces more Pierpont numbers than it needs to fulfill the requested number of primes. It uses a series of parallel iterators with additional iterators added as required. No need to speculatively generate an overabundance. No need to rely on magic numbers. No need to sort them. It produces exactly what is needed, in order, on demand.
use ntheory:from<Perl5> <is_prime>;
sub pierpont ($kind is copy = 1) {
fail "Unknown type: $kind. Must be one of 1 (default) or 2" if $kind !== 1|2;
$kind = -1 if $kind == 2;
my $po3 = 3;
my $add-one = 3;
my @iterators = [1,2,4,8 … *].iterator, [3,9,27 … *].iterator;
my @head = @iterators».pull-one;
gather {
loop {
my $key = @head.pairs.min( *.value ).key;
my $min = @head[$key];
@head[$key] = @iterators[$key].pull-one;
take $min + $kind if "{$min + $kind}".&is_prime;
if $min >= $add-one {
@iterators.push: ([|((2,4,8).map: * * $po3) … *]).iterator;
$add-one = @head[+@iterators - 1] = @iterators[+@iterators - 1].pull-one;
$po3 *= 3;
}
}
}
}
say "First 50 Pierpont primes of the first kind:\n" ~ pierpont[^50].rotor(10)».fmt('%8d').join: "\n";
say "\nFirst 50 Pierpont primes of the second kind:\n" ~ pierpont(2)[^50].rotor(10)».fmt('%8d').join: "\n";
say "\n250th Pierpont prime of the first kind: " ~ pierpont[249];
say "\n250th Pierpont prime of the second kind: " ~ pierpont(2)[249];
Output:
First 50 Pierpont primes of the first kind:
2 3 5 7 13 17 19 37 73 97
109 163 193 257 433 487 577 769 1153 1297
1459 2593 2917 3457 3889 10369 12289 17497 18433 39367
52489 65537 139969 147457 209953 331777 472393 629857 746497 786433
839809 995329 1179649 1492993 1769473 1990657 2654209 5038849 5308417 8503057
First 50 Pierpont primes of the second kind:
2 3 5 7 11 17 23 31 47 53
71 107 127 191 383 431 647 863 971 1151
2591 4373 6143 6911 8191 8747 13121 15551 23327 27647
62207 73727 131071 139967 165887 294911 314927 442367 472391 497663
524287 786431 995327 1062881 2519423 10616831 17915903 18874367 25509167 30233087
250th Pierpont prime of the first kind: 62518864539857068333550694039553
250th Pierpont prime of the second kind: 4111131172000956525894875083702271
Generalized Pierpont iterator
Alternately, a version that will generate generalized Pierpont numbers for any set of prime integers where at least one of the primes is 2.
(Cut down output as it is exactly the same as the first version for {2,3} +1 and {2,3} -1; leaves room to demo some other options.)
sub smooth-numbers (*@list) {
cache my \Smooth := gather {
my %i = (flat @list) Z=> (Smooth.iterator for ^@list);
my %n = (flat @list) Z=> 1 xx *;
loop {
take my $n := %n{*}.min;
for @list -> \k {
%n{k} = %i{k}.pull-one * k if %n{k} == $n;
}
}
}
}
# Testing various smooth numbers
for 'OEIS: A092506 - 2 + Fermat primes:', (2), 1, 6,
"\nOEIS: A000668 - Mersenne primes:", (2), -1, 10,
"\nOEIS: A005109 - Pierpont primes 1st:", (2,3), 1, 20,
"\nOEIS: A005105 - Pierpont primes 2nd:", (2,3), -1, 20,
"\nOEIS: A077497:", (2,5), 1, 20,
"\nOEIS: A077313:", (2,5), -1, 20,
"\nOEIS: A002200 - (\"Hamming\" primes 1st):", (2,3,5), 1, 20,
"\nOEIS: A293194 - (\"Hamming\" primes 2nd):", (2,3,5), -1, 20,
"\nOEIS: A077498:", (2,7), 1, 20,
"\nOEIS: A077314:", (2,7), -1, 20,
"\nOEIS: A174144 - (\"Humble\" primes 1st):", (2,3,5,7), 1, 20,
"\nOEIS: A299171 - (\"Humble\" primes 2nd):", (2,3,5,7), -1, 20,
"\nOEIS: A077499:", (2,11), 1, 20,
"\nOEIS: A077315:", (2,11), -1, 20,
"\nOEIS: A173236:", (2,13), 1, 20,
"\nOEIS: A173062:", (2,13), -1, 20
-> $title, $primes, $add, $count {
say "$title smooth \{$primes\} {$add > 0 ?? '+' !! '-'} 1 ";
put smooth-numbers(|$primes).map( * + $add ).grep( *.is-prime )[^$count]
}
Output:
OEIS: A092506 - 2 + Fermat primes: smooth {2} + 1
2 3 5 17 257 65537
OEIS: A000668 - Mersenne primes: smooth {2} - 1
3 7 31 127 8191 131071 524287 2147483647 2305843009213693951 618970019642690137449562111
OEIS: A005109 - Pierpont primes 1st: smooth {2 3} + 1
2 3 5 7 13 17 19 37 73 97 109 163 193 257 433 487 577 769 1153 1297
OEIS: A005105 - Pierpont primes 2nd: smooth {2 3} - 1
2 3 5 7 11 17 23 31 47 53 71 107 127 191 383 431 647 863 971 1151
OEIS: A077497: smooth {2 5} + 1
2 3 5 11 17 41 101 251 257 401 641 1601 4001 16001 25601 40961 62501 65537 160001 163841
OEIS: A077313: smooth {2 5} - 1
3 7 19 31 79 127 199 499 1249 1279 1999 4999 5119 8191 12799 20479 31249 49999 51199 79999
OEIS: A002200 - ("Hamming" primes 1st): smooth {2 3 5} + 1
2 3 5 7 11 13 17 19 31 37 41 61 73 97 101 109 151 163 181 193
OEIS: A293194 - ("Hamming" primes 2nd): smooth {2 3 5} - 1
2 3 5 7 11 17 19 23 29 31 47 53 59 71 79 89 107 127 149 179
OEIS: A077498: smooth {2 7} + 1
2 3 5 17 29 113 197 257 449 1373 3137 50177 65537 114689 268913 470597 614657 1075649 3294173 7340033
OEIS: A077314: smooth {2 7} - 1
3 7 13 31 97 127 223 1567 3583 4801 6271 8191 19207 25087 33613 76831 131071 401407 524287 917503
OEIS: A174144 - ("Humble" primes 1st): smooth {2 3 5 7} + 1
2 3 5 7 11 13 17 19 29 31 37 41 43 61 71 73 97 101 109 113
OEIS: A299171 - ("Humble" primes 2nd): smooth {2 3 5 7} - 1
2 3 5 7 11 13 17 19 23 29 31 41 47 53 59 71 79 83 89 97
OEIS: A077499: smooth {2 11} + 1
2 3 5 17 23 89 257 353 1409 2663 30977 65537 170369 495617 5767169 23068673 59969537 82458113 453519617 3429742097
OEIS: A077315: smooth {2 11} - 1
3 7 31 43 127 241 967 5323 8191 117127 131071 524287 7496191 10307263 77948683 253755391 428717761 738197503 1714871047 2147483647
OEIS: A173236: smooth {2 13} + 1
2 3 5 17 53 257 677 3329 13313 35153 65537 2768897 13631489 2303721473 3489660929 4942652417 11341398017 10859007357953 1594691292233729 31403151600910337
OEIS: A173062: smooth {2 13} - 1
3 7 31 103 127 337 1663 5407 8191 131071 346111 524287 2970343 3655807 22151167 109051903 617831551 1631461441 2007952543 2147483647