carmichael_with_3_factors_p+1|n+1.pl

#!/usr/bin/perl

# Carmichael numbers m that have at least 3 prime factors p such that p+1 | m+1.
# https://oeis.org/A329948

# It is not known whether any Carmichael number (A002997) is also a Lucas-Carmichael number (A006972).

use 5.020;
use strict;
use warnings;

use ntheory qw(:all);
use Math::Prime::Util::GMP;
use Math::GMPz;

while (<>) {
    next if /^\h*#/;
    /\S/ or next;
    my $n = (split(' ', $_))[-1];

    $n || next;

    next if ($n < ~0);
    next if length($n) > 35;

    #is_smooth($n, 1e5) || next;

    Math::Prime::Util::GMP::is_carmichael($n) || next;
    $n = Math::GMPz->new($n);

    my $inc = $n + 1;
    my $k   = scalar grep { $inc % ($_ + 1) == 0 } Math::Prime::Util::GMP::factor($n);

    if ($k >= 3) {
        say "$k: $n";
    }
}

__END__

# Terms below 2^64:

9857524690572481
33439671284716801
96653613831890401
270136961300544031
528096456788419441
650643395658753601
710238404427321601
1822922951416158241
4011563714063821201
4525693104167627041
4631812281009523441
7049793086137296001
8605736094003523201
10449416165574628801
11175581620177915681
12746447178170148001
12769123623410580481
17705945296667070001

# Extra terms:

112063204636898630401
171864347319293434273
290511032565049956001
2433475541834218396801
5500350467752540752001
15766720433614895241601
702716905099325753679601
1641197929815383993299441
1694870871860587162176721
2352002136477575651201920801
12939435103076881117129002241
16921253986821645965065105921
100316367094716219646271183761
123702372517731523759641748755329
379305888565700146110687208821889
21424389954178567168694716161869094785
1267302465895414494311727362263836874081
6705723249444927031929382532702808317638047551685696855129161734440001