prog.pl

#!/usr/bin/perl

# Carmichael numbers k such that k+1 is divisible by gpf(k)+1, where gpf = A006530.
# https://oeis.org/A335336

# Find Carmichael numbers m such that gpf(m)+1 divides m+1, where gpf(n) is the greatest prime factor of n.

# isok(n) = my(f=factor(n));

#Richard J.McIntosh and Mitra Dipra, <a href="https://www.sciencedirect.com/science/article/pii/S0022314X14002108">Carmichael numbers with p+1|n+1</a>, Journal of Number Theory, Volume 147, February 2015, Pages 81-91.

# Are there any Carmichael numbers k with exactly four prime factors such that gpf(k)+1 divides k+1?

use 5.020;
use strict;
use warnings;

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

#my %table;

while (<>) {

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

    #$n || next;

   # next if ($n < ~0);
    #next if length($n) > 30;
    #next if length($n) <= 30;

    #is_smooth($n, 1e5) || next;
    #Math::Prime::Util::GMP::is_carmichael($n) || next;
    #$n = Math::GMPz->new($n);

    $n = Math::GMPz->new($n);

    my $np1 = $n+1;
    my $pp1 = ((factor($n))[-1])+1;

    if ($np1 % $pp1 == 0) {
        say "$n";
    }
}

__END__
687979968481
1928376089641
2638625591701
3148470889201
3152088903601
14682521533681
19656816822721
37333372057201
47003559452641
80643055074121
129235662445121
140940741166849
196945133626801
336301807660741
345186571310209
439931062854361
490464498901105
531259716925081
707472049888801
735691828435201
778963619135521
788801817683881
893955162830521
1464274069272001
2251648591728481
3908308008236941
5329243598526001
5581318384852561
13326785007243841
33974062308537121
34032752544117601
54853966343767501
68134265628726529
110142636908589649
114346331280789301
258874196100780961
364059505086209101
380775595507169941
627103832138305831
875815255956300661
905669264186473501
908953689898563121
1309513581297673921
1403552667045816001
1777998645730488061
1965645625719291121
4564042295765073121
8605736094003523201
14666471748992794501
17907650644750888141

Several terms greater than 2^64:

27453375098872012081
1553120766503217777601
2316867089206526090641
5745765587660958183661
170845977240460180827001
211376155831990085740609
9920799542911938327843841
375623792044867850082496129
15853017707070075884775600001