```
# 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::HappySteps;
use 5.004;
use strict;
use vars '$VERSION', '@ISA';
$VERSION = 72;
use Math::NumSeq;
use Math::NumSeq::Base::IterateIth;
@ISA = ('Math::NumSeq::Base::IterateIth',
'Math::NumSeq');
*_is_infinite = \&Math::NumSeq::_is_infinite;
# uncomment this to run the ### lines
#use Devel::Comments;
# use constant name => Math::NumSeq::__('Happy Steps');
use constant description => Math::NumSeq::__('How many sum of squares of digits steps to get to a repeating iteration.');
use constant i_start => 1;
use constant values_min => 1;
use constant characteristic_count => 1;
use constant characteristic_smaller => 1;
use constant characteristic_increasing => 0;
use Math::NumSeq::Base::Digits
'parameter_info_array'; # radix parameter
#------------------------------------------------------------------------------
# cf A001273 smallest happy which takes N steps
#
my @oeis_anum;
$oeis_anum[2] = 'A078627'; # starting i=1 ...
# OEIS-Catalogue: A078627 radix=2
$oeis_anum[10] = 'A193995'; # but starting i=1 ...
# OEIS-Catalogue: A193995
sub oeis_anum {
my ($self) = @_;
return $oeis_anum[$self->{'radix'}];
}
#------------------------------------------------------------------------------
sub ith {
my ($self, $i) = @_;
if ($i <= 0) {
return 0;
}
if (_is_infinite($i)) {
return $i;
}
my $radix = $self->{'radix'};
my $steps = 0;
my %seen;
for (;;) {
### $i
my $sum = 0;
if ($seen{$i}) {
return $steps;
}
$seen{$i} = 1;
while ($i) {
my $digit = ($i % $radix);
$sum += $digit * $digit;
$i = int($i/$radix);
}
$i = $sum;
$steps++;
}
}
sub pred {
my ($self, $value) = @_;
### HappySteps pred(): $value
return ($value >= 0 && $value == int($value));
}
1;
__END__
=for stopwords Ryde HappyNumbers HappySteps Math-NumSeq Radix
=head1 NAME
Math::NumSeq::HappySteps -- number of sum of squares of digits iterations to reach a repeat
=head1 SYNOPSIS
use Math::NumSeq::HappySteps;
my $seq = Math::NumSeq::HappySteps->new (radix => 10);
my ($i, $value) = $seq->next;
=head1 DESCRIPTION
This is the number of iterations of the C<HappyNumbers> style "sum of
squares of digits" is required to reach a repeat of a value seen before, and
therefore to establish whether a number is a happy number or not.
1, 9, 13, 8, 12, 17, 6, 13, 12, 2,
starting i=1
For example i=10 is value 2 because 10-E<gt>1-E<gt>1 is 2 iterations to get
to a repeat (a repeat of 1). At i=1 itself the value is 1 since 1 iteration
reaches 1 again which is itself the repeat. That count 1 at i=1 is the
minimum.
=head2 Radix
An optional C<radix> parameter selects a base other than decimal. In binary
C<radix=E<gt>2> the digits are all either 0 or 1 so "sum of squares of
digits" is the same as a plain "sum of digits".
In some bases there's longer cycles than others which a non-happy number
might fall into. For example base 20 has a cycle
10 -> 100 -> 25 -> 26 -> ... -> 61 -> 10
total 26 elements
When a non-happy falls into such a cycle its C<HappySteps> count here is at
least 26 (or whatever amount) to reach a repeat.
=head1 FUNCTIONS
See L<Math::NumSeq/FUNCTIONS> for behaviour common to all sequence classes.
=over 4
=item C<$seq = Math::NumSeq::HappySteps-E<gt>new ()>
=item C<$seq = Math::NumSeq::HappySteps-E<gt>new (radix =E<gt> $r)>
Create and return a new sequence object.
=back
=head2 Random Access
=over
=item C<$value = $seq-E<gt>ith($i)>
Return the number of iterations starting from C<$i> required to reach a
repeat.
=back
=head1 SEE ALSO
L<Math::NumSeq>,
L<Math::NumSeq::HappyNumbers>,
L<Math::NumSeq::DigitSum>
=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
```