# Copyright 2011, 2012, 2013, 2014 Kevin Ryde
# This file is part of Math-NumSeq.
#
# Math-NumSeq is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 3, or (at your option) any later
# version.
#
# Math-NumSeq is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
# or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
# for more details.
#
# You should have received a copy of the GNU General Public License along
# with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.
package Math::NumSeq::DivisorCount;
use 5.004;
use strict;
use vars '$VERSION','@ISA';
$VERSION = 71;
use Math::NumSeq;
use Math::NumSeq::Base::IterateIth;
@ISA = ('Math::NumSeq::Base::IterateIth',
'Math::NumSeq');
use Math::NumSeq::PrimeFactorCount;
*_prime_factors = \&Math::NumSeq::PrimeFactorCount::_prime_factors;
# uncomment this to run the ### lines
#use Devel::Comments;
# use constant name => Math::NumSeq::__('Divisor Count');
use constant description => Math::NumSeq::__('Count of divisors of i (including 1 and i).');
use constant i_start => 1;
use constant characteristic_count => 1;
use constant characteristic_smaller => 1;
use constant characteristic_increasing => 0;
# "proper" divisors just means 1 less in each value, not sure much use for
# that.
#
# n itself -- proper, or not
# 1 -- proper, or not
# square, non-square
#
# use constant parameter_info_array =>
# [ { name => 'divisor_type',
# display => Math::NumSeq::__('Divisor Type'),
# type => 'enum',
# choices => ['all','proper'], # ,'propn1'
# default => 'all',
# # description => Math::NumSeq::__(''),
# },
# ];
my %values_min = (all => 1,
proper => 0,
propn1 => 0);
sub values_min {
my ($self) = @_;
# or values_min=0 if i_start=0
return 1; # $values_min{$self->{'divisor_type'}};
}
#------------------------------------------------------------------------------
# cf A032741 - 1 <= d < n starting n=0
# A147588 - 1 < d < n starting n=1
#
# A006218 - cumulative count of divisors
# A002541 - cumulative proper divisors
#
# A001227 - count odd divisors
# A001826 - count 4k+1 divisors
# A038548 - count divisors <= sqrt(n)
# A070824 - proper divisors starting n=2
# A002182 - number with new highest number of divisors
# A002183 - that count of divisors
# A001876 - count 5k+1 divisors
# A001877 - count 5k+2 divisors
# A001878 - count 5k+3 divisors
# A001899 - count 5k+4 divisors
#
# A028422 - count of ways to factorize
# A033834 - n with new high count factorizations
# A033833 - highly factorable
#
# A056595 - count non-square divisors
# A046951 - count square divisors
# A013936 - cumulative count square divisors
# A137518 - same divisor count as n, and > a(n-1) so increasing
#
sub oeis_anum {
my ($self) = @_;
return 'A000005';
# OEIS-Catalogue: A000005
# my %oeis_anum = (all => 'A000005', # all divisors starting n=1
# # proper => 'A032741', # starts n=0 ...
# # propn1 => 'A147588',
# );
# return $oeis_anum{$self->{'divisor_type'}};
}
#------------------------------------------------------------------------------
sub ith {
my ($self, $i) = @_;
$i = abs($i);
if ($i == 0) {
return 0;
}
# If i = p^a * q^b * ... then divisorcount = (a+1)*(b+1)*...
# which is each possible power p^0, p^1, ..., p^a of each prime,
# including all zeros p^0*q^0*... = 1 and p^a*q^b*... itself.
#
# If i is a primorial 2*3*5*7*13*... with k primes then divisorcount=2^k
# so the $value product can become a bignum if $i is a bignum.
my ($good, @primes) = _prime_factors($i);
return undef unless $good;
my $value = ($i*0) + 1; # inherit possible bignum
my $prev = 0;
my $dcount = 1;
while (my $p = shift @primes) {
if ($p == $prev) {
$dcount++;
} else {
$value *= $dcount;
$dcount = 2;
$prev = $p;
}
}
return $value * $dcount;
# if ($self->{'divisor_type'} eq 'propn1') {
# if ($ret <= 2) {
# return 0;
# }
# $ret -= 2;
# }
}
sub pred {
my ($self, $value) = @_;
return ($value >= 1 && $value == int($value));
}
1;
__END__
# This was next() done by sieve, but it's slower from about i=5000 with XS
# code for prime_factors() and it uses a lot of memory if continue next()
# for a long time.
#
# sub rewind {
# my ($self) = @_;
# ### DivisorCount rewind()
# $self->{'i'} = $self->i_start;
# _restart_sieve ($self, 5);
# }
# sub _restart_sieve {
# my ($self, $hi) = @_;
#
# $self->{'hi'} = $hi;
# $self->{'array'} = [ 0, (1) x $self->{'hi'} ];
# }
#
# sub next {
# my ($self) = @_;
# ### DivisorCount next(): $self->{'i'}
#
# my $hi = $self->{'hi'};
# my $start = my $i = $self->{'i'}++;
# if ($i > $hi) {
# _restart_sieve ($self, $hi *= 2);
# $start = 2;
# }
#
# my $aref = $self->{'array'};
# if ($start <= $i) {
# if ($start < 2) {
# $start = 2;
# }
# foreach my $i ($start .. $i) {
# if ($aref->[$i] == 1) {
# ### apply prime: $i
# my $step = 1;
# for (my $pcount = 1; ; $pcount++) {
# $step *= $i;
# ### $step
# last if ($step > $hi);
# my $pmul = $pcount+1;
# for (my $j = $step; $j <= $hi; $j += $step) {
# ($aref->[$j] /= $pcount) *= $pmul;
# }
# # last if $self->{'divisor_type'} eq 'propn1';
# }
# # print "applied: $i\n";
# # for (my $j = 0; $j < $hi; $j++) {
# # printf " %2d %2d\n", $j, vec($$aref, $j,8));
# # }
# }
# }
# }
# ### ret: "$i, $aref->[$i]"
# return ($i, $aref->[$i]);
# }
=for stopwords Ryde Math-NumSeq
=head1 NAME
Math::NumSeq::DivisorCount -- how many divisors
=head1 SYNOPSIS
use Math::NumSeq::DivisorCount;
my $seq = Math::NumSeq::DivisorCount->new;
my ($i, $value) = $seq->next;
=head1 DESCRIPTION
The number of divisors of i,
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, ...
starting i=1
i=1 is divisible only by 1 so value=1. Then i=2 is divisible by 1 and 2 so
value=2. Or for example i=6 is divisible by 4 numbers 1,2,3,6 so value=4.
=head1 FUNCTIONS
See L<Math::NumSeq/FUNCTIONS> for behaviour common to all sequence classes.
=over 4
=item C<$seq = Math::NumSeq::DivisorCount-E<gt>new ()>
Create and return a new sequence object.
=back
=head2 Random Access
=over
=item C<$value = $seq-E<gt>ith($i)>
Return the number of prime factors in C<$i>.
This calculation requires factorizing C<$i> and in the current code after
small factors a hard limit of 2**32 is enforced in the interests of not
going into a near-infinite loop.
=item C<$bool = $seq-E<gt>pred($value)>
Return true if C<$value> occurs as a divisor count, which simply means
C<$value E<gt>= 1>.
=back
=head1 SEE ALSO
L<Math::NumSeq>,
L<Math::NumSeq::PrimeFactorCount>
L<Math::Factor::XS>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-numseq/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
Math-NumSeq is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
more details.
You should have received a copy of the GNU General Public License along with
Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.
=cut