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- #!/usr/bin/perl
- # Daniel "Trizen" Șuteu
- # Date: 11 January 2019
- # https://github.com/trizen
- # Fast program for computing the numbers `n` such that the denominator of Bernoulli(n) is a record.
- # OEIS sequences:
- # https://oeis.org/A100195
- # https://oeis.org/A100194
- # See also:
- # https://en.wikipedia.org/wiki/Bernoulli_number
- # https://mathworld.wolfram.com/BernoulliNumber.html
- # https://en.wikipedia.org/wiki/Von_Staudt%E2%80%93Clausen_theorem
- use 5.020;
- use strict;
- use warnings;
- use experimental qw(signatures);
- use ntheory qw(divisors is_prime vecprod);
- sub bernoulli_denominator ($n) { # denominator of the n-th Bernoulli number
- return 1 if ($n <= 0);
- return 1 if ($n > 1 and $n % 2);
- vecprod(map { $_ + 1 } grep { is_prime($_ + 1) } divisors($n));
- }
- sub records_upto ($n, $callback) {
- for (my ($k, $m) = (0, -1) ; $k <= $n ; $k += 2) {
- my $sum = 0;
- foreach my $d (divisors($k)) {
- if (is_prime($d + 1)) {
- $sum += log($d + 1);
- }
- }
- if ($sum > $m) {
- $m = $sum;
- $callback->($k);
- }
- }
- }
- records_upto(1e4, sub ($k) { say "B($k) = ", bernoulli_denominator($k) });
- __END__
- B(0) = 2
- B(2) = 6
- B(4) = 30
- B(6) = 42
- B(10) = 66
- B(12) = 2730
- B(30) = 14322
- B(36) = 1919190
- B(60) = 56786730
- B(72) = 140100870
- B(108) = 209191710
- B(120) = 2328255930
- B(144) = 2381714790
- B(180) = 7225713885390
- B(240) = 9538864545210
- B(360) = 21626561658972270
- B(420) = 446617991732222310
- B(540) = 115471236091149548610
- B(840) = 5145485882746933233510
- B(1008) = 14493038256293268734790
- B(1080) = 345605409620810598989730
- B(1200) = 42107247672297314156359710
- B(1260) = 4554106624556364764691012210
- B(1620) = 24743736851520275624910204330
- B(1680) = 802787680649929796414310788070
- B(2016) = 1908324101335116127448341021830
- B(2160) = 1324918483651364394207119201026530
- B(2520) = 9655818125018463593525930077544596530
- B(3360) = 176139196253087613320507734410708168870
- B(3780) = 20880040554948303778681975110988542692370
- B(5040) = 1520038371910163024272084596792024938493098335890
- B(6480) = 2386506545702609292996755910476726098859145077130
- B(7560) = 334731403390662540713247087231623394273840419057927010
- B(8400) = 30721852291400450355987797336504062619723310330260297070
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