abundancy_index_records_psp2.pl

#!/usr/bin/perl

# Poulet numbers (Fermat pseudoprimes to base 2) k that have an abundancy index sigma(k)/k that is larger than the abundancy index of all smaller Poulet numbers.
# https://oeis.org/A328691

# Known terms:
#   341, 561, 645, 18705, 2113665, 2882265, 81722145, 9234602385, 19154790699045, 43913624518905, 56123513337585, 162522591775545, 221776809518265, 3274782926266545, 4788772759754985

# Several abundant Fermat pseudoprimes to base 2:
#   898943937249247967890084629421065
#   222042825169546323981793629414604065
#   2596282479202818734176082185090403265
#   12796625128232187655293894634808130945
#   3470207934739664512679701940114447720865

use 5.020;
use strict;
use warnings;

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

use Math::GMPz;
use Math::GMPq;
use Math::MPFR;

my %seen;
my $max = 0;

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

    $n || next;
    $n =~ /^\d+\z/ or next;

    #next if $n < ~0;
    #next if length($n) > 65;

    $n = Math::GMPz->new($n);
    $n % (3 * 5 * 17 * 23 * 29 * 43 * 53 * 89) == 0 or next;

    Math::Prime::Util::GMP::is_pseudoprime($n, 2) || next;

    #say "Testing: $n";

    my $t     = Math::GMPq->new(0);
    my $sigma = divisor_sum($n);
    Math::GMPq::Rmpq_set_str($t, "$sigma", 10);
    Math::GMPq::Rmpq_div_z($t, $t, $n);

    if ($t > $max) {

        #if ($t >= 2) {
        #if ($t >= 1.9) {

        #die "Found: $n";
        #say $n;

        printf "%.4f abundancy: $n\n", Math::MPFR->new($t) if !$seen{$n}++;
        $max = $t;
    }
}

__END__
1.1261 abundancy: 341
1.5401 abundancy: 561
1.6372 abundancy: 645
1.6937 abundancy: 18705
1.7087 abundancy: 2113665
1.7266 abundancy: 2882265
1.8003 abundancy: 81722145
1.8162 abundancy: 9234602385
1.8211 abundancy: 19154790699045
1.8236 abundancy: 43913624518905
1.8459 abundancy: 56123513337585
1.8632 abundancy: 162522591775545
1.9036 abundancy: 221776809518265
1.9106 abundancy: 3274782926266545
1.9449 abundancy: 4788772759754985
1.9452 abundancy: 38353281032877674865
1.9452 abundancy: 2638294879881771254145
1.9454 abundancy: 24773130788808935997465
1.9454 abundancy: 429388072625313108651345
1.9459 abundancy: 633708839387221385771985
1.9594 abundancy: 4426632427184293146004185
1.9786 abundancy: 30289748358904212366572385
1.9814 abundancy: 728017010426459878356936705
1.9842 abundancy: 34742187522900518220477982065
1.9856 abundancy: 296874628838164374175009735905
1.9888 abundancy: 282385466973939051351462060010785
1.9893 abundancy: 303874597856107607075523212740305
1.9967 abundancy: 95232935093163427870007414104051665
2.0051 abundancy: 2596282479202818734176082185090403265
2.0020 abundancy: 12796625128232187655293894634808130945
2.0027 abundancy: 3470207934739664512679701940114447720865
2.0027 abundancy: 35681625351759414447784656906886402710161726145
2.0036 abundancy: 18324199296528369643710689094829325758885303308048705
2.0036 abundancy: 3521782835397676946933688469202532605577443098683804760065
2.0075 abundancy: 4480869812756224982980361246336523683775981809514868335105

# New data (10 March 2023):

1.9449 abundancy: 4788772759754985
1.9452 abundancy: 38353281032877674865
1.9454 abundancy: 24773130788808935997465
1.9454 abundancy: 429388072625313108651345
1.9459 abundancy: 633708839387221385771985
1.9594 abundancy: 4426632427184293146004185
1.9704 abundancy: 5483074269820204877403585
1.9786 abundancy: 30289748358904212366572385
1.9814 abundancy: 728017010426459878356936705
1.9842 abundancy: 34742187522900518220477982065
1.9851 abundancy: 105741753586113830661567555465
1.9869 abundancy: 112558094252473932795411783105
1.9913 abundancy: 7935079227520111979253186509265
1.9971 abundancy: 78494365937003386374263317442385
2.0013 abundancy: 898943937249247967890084629421065
2.0040 abundancy: 222042825169546323981793629414604065
2.0052 abundancy: 2596282479202818734176082185090403265
2.0075 abundancy: 4480869812756224982980361246336523683775981809514868335105