# Copyright 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::DedekindPsiSteps;
use 5.004;
use strict;
use vars '$VERSION', '@ISA';
$VERSION = 69;
use Math::NumSeq;
use Math::NumSeq::Base::IterateIth;
@ISA = ('Math::NumSeq::Base::IterateIth',
'Math::NumSeq');
*_is_infinite = \&Math::NumSeq::_is_infinite;
use Math::NumSeq::PrimeFactorCount;;
*_prime_factors = \&Math::NumSeq::PrimeFactorCount::_prime_factors;
# uncomment this to run the ### lines
#use Smart::Comments;
# use constant name => Math::NumSeq::__('Dedekind Psi Steps');
use constant description => Math::NumSeq::__('Number of repeated applications of the Dedekind psi function to reach factors 2 and 3 only.');
use constant default_i_start => 1;
use constant characteristic_count => 1;
use constant characteristic_smaller => 1;
use constant characteristic_increasing => 0;
use constant values_min => 0;
# A001615 dedekind psi
# A019268 dedekind psi first requiring n steps
# A019269 number of steps
# A173290 dedekind psi cumulative
#
use constant oeis_anum => 'A019269';
sub ith {
my ($self, $i) = @_;
### DedekindPsiSteps ith(): $i
if (_is_infinite($i)) {
return $i;
}
if ($i < 0) {
return undef;
}
my ($good, @primes) = _prime_factors($i);
return undef unless $good;
my %primes;
foreach my $p (@primes) {
$primes{$p}++;
}
my $count = 0;
my %prime_factors;
for (;;) {
delete $primes{'2'};
delete $primes{'3'};
last unless %primes;
### %primes
$count++;
my %next_primes;
while (my ($p, $e) = each %primes) {
if (--$e) {
$next_primes{$p} += $e;
}
my $prime_factors_aref = ($prime_factors{$p} ||= do {
my ($good, @primes) = _prime_factors($p+1);
return undef unless $good;
\@primes
});
foreach my $f (@$prime_factors_aref) {
$next_primes{$f}++;
}
}
%primes = %next_primes;
}
return $count;
}
# sub _psi_ex23 {
# my ($n) = @_;
# ### _psi(): $n
#
# my $prev = 0;
# my $ret = 1;
# foreach my $p (prime_factors($n)) {
# next if $p == 2 || $p == 3;
# if ($p == $prev) {
# $ret *= $p;
# } else {
# $ret *= $p+1;
# $prev = $p;
# }
# }
# return $ret;
#
# # my %primes;
# # @primes{prime_factors($n)} = (); # hash slice
# # delete $primes{2,3}; # hash slice
# # foreach my $p (keys %primes) {
# # $n /= $p;
# # $n *= $p+1;
# # }
# return $n;
# }
# sub _is_2x3y {
# my ($n) = @_;
# until ($n % 2) {
# $n = int($n/2);
# }
# until ($n % 3) {
# $n = int($n/3);
# }
# return ($n == 1);
# }
1;
__END__
=for stopwords Ryde Math-NumSeq ie
=head1 NAME
Math::NumSeq::DedekindPsiSteps -- psi function until 2^x*3^y
=head1 SYNOPSIS
use Math::NumSeq::DedekindPsiSteps;
my $seq = Math::NumSeq::DedekindPsiSteps->new;
my ($i, $value) = $seq->next;
=head1 DESCRIPTION
This sequence is the how many repeated applications of the Dedekind psi
function are required to reach a number of the form 2^x*3^y.
0,0,0,0,1,0,1,0,0,1,1,0,2,1,1,0,1,0,2,1,1,1,1,0,2,2,0,1,2,...
The psi function is
psi(n) = product (p+1) * p^(e-1)
prime factors p^e in n
The p+1 means that one copy of each distinct prime in n is changed from p to
p+1. That p+1 is even, so although the value has increased the prime
factors are all less than p. Repeated applying that reduction eventually
reaches just primes 2 and 3 in some quantity.
For example i=25 requires 2 steps,
psi(25) = (5+1)*5 = 30 = 2*3*5
then
psi(30) = (2+1)*(3+1)*(5+1) = 72 = 2*2*2*3*3
If i is already 2s and 3s then it's considered no steps are required and the
value is 0. For example at i=12=2*2*3 the value is 0.
=head1 FUNCTIONS
See L<Math::NumSeq/FUNCTIONS> for behaviour common to all sequence classes.
=over 4
=item C<$seq = Math::NumSeq::DedekindPsiSteps-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 repeated applications of the psi function on C<$i>
required to reach just factors 2 and 3.
This calculation requires factorizing C<$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. Above that the return is C<undef>.
=back
=head1 SEE ALSO
L<Math::NumSeq>,
L<Math::NumSeq::TotientSteps>,
L<Math::NumSeq::DedekindPsiCumulative>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-numseq/index.html>
=head1 LICENSE
Copyright 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