lib/Math/NumSeq/RadixWithoutDigit.pm

# Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 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::RadixWithoutDigit;
use 5.004;
use strict;

use vars '$VERSION', '@ISA';
$VERSION = 75;

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 Smart::Comments;

# use constant name => Math::NumSeq::__('Without chosen digit');
use constant description => Math::NumSeq::__('Integers which don\'t have a given digit when written out in a radix.');
use constant default_i_start => 1;
use constant characteristic_increasing => 1;
use constant characteristic_integer => 1;

use Math::NumSeq::Base::Digits;
use constant parameter_info_array =>
  [ Math::NumSeq::Base::Digits->parameter_info_list(),
    {
     name    => 'digit',
     type    => 'integer',
     display => Math::NumSeq::__('Digit'),
     default => -1,
     minimum => -1,
     width   => 2,
     description => Math::NumSeq::__('Digit to exclude.  Default -1 means the highest digit, ie. radix-1.'),
    },
  ];

#------------------------------------------------------------------------------
# cf A011531 - numbers with a 1 digit
#    A011532 ... A011539 - numbers with a 2 to 9 digit
#

my @oeis_anum;

# Not quite A000225 = 2^n-1 is base 2 without 0, but includes a(0)=0 as
# 2^0-1 = 0 whereas RadixWithoutDigit doesn't include value=0.
# A000225 is generated by Math::NumSeq::Repdigits anyway.

$oeis_anum[1]->[3]->[0] = 'A032924'; # base 3 no 0    OFFSET=1
$oeis_anum[1]->[3]->[1] = 'A005823'; # base 3 no 1    OFFSET=1
$oeis_anum[1]->[3]->[2] = 'A005836'; # base 3 no 2    OFFSET=1
# OEIS-Catalogue: A032924 radix=3 digit=0  # base 3 no 0
# OEIS-Catalogue: A005823 radix=3 digit=1  # base 3 no 1
# OEIS-Catalogue: A005836 radix=3 digit=2  # base 3 no 2

$oeis_anum[1]->[4]->[0] = 'A023705'; # base 4 no 0    OFFSET=1
$oeis_anum[1]->[4]->[1] = 'A023709'; # base 4 no 1    OFFSET=1
$oeis_anum[1]->[4]->[2] = 'A023713'; # base 4 no 2    OFFSET=1
$oeis_anum[0]->[4]->[3] = 'A023717'; # base 4 no 3    OFFSET=0
# OEIS-Catalogue: A023705 radix=4 digit=0  # base 4 no 0
# OEIS-Catalogue: A023709 radix=4 digit=1  # base 4 no 1
# OEIS-Catalogue: A023713 radix=4 digit=2  # base 4 no 2
# OEIS-Catalogue: A023717 radix=4 digit=3 i_start=0  # base 4 no 3

$oeis_anum[1]->[5]->[0] = 'A023721'; # base 5 no 0    OFFSET=1
$oeis_anum[1]->[5]->[1] = 'A023725'; # base 5 no 1    OFFSET=1
$oeis_anum[1]->[5]->[2] = 'A023729'; # base 5 no 2    OFFSET=1
$oeis_anum[1]->[5]->[3] = 'A023733'; # base 5 no 3    OFFSET=1
$oeis_anum[1]->[5]->[4] = 'A023737'; # base 5 no 4    OFFSET=1
# OEIS-Catalogue: A023721 radix=5 digit=0  # base 5 no 0
# OEIS-Catalogue: A023725 radix=5 digit=1  # base 5 no 1
# OEIS-Catalogue: A023729 radix=5 digit=2  # base 5 no 2
# OEIS-Catalogue: A023733 radix=5 digit=3  # base 5 no 3
# OEIS-Catalogue: A023737 radix=5 digit=4  # base 5 no 4

# cf A037465 radix=6 digit=5 i_start=1 # base 6 no 5, starting i=1

$oeis_anum[1]->[7]->[6] = 'A020657'; # "no 7-term arithmetic progression" OFFSET=1
# OEIS-Catalogue: A020657 radix=7 digit=6

# starting OFFSET=1 value=1
# $oeis_anum[1]->[8]->[7] = 'A037474'; # base 7 written out in octal
# # OEIS-Catalogue: A037474 radix=8 digit=7 i_start=1

# starting OFFSET=1 value=1
# $oeis_anum[1]->[9]->[8] = 'A037477'; # base 8 written out in base 9
# # OEIS-Catalogue: A037477 radix=9 digit=8 i_start=1

$oeis_anum[1]->[10]->[0] = 'A052382'; # decimal numbers without 0, OFFSET=1
$oeis_anum[1]->[10]->[1] = 'A052383'; # decimal numbers without 1, OFFSET=1 start 0
$oeis_anum[0]->[10]->[2] = 'A052404'; # decimal numbers without 2, OFFSET=0 start 0
$oeis_anum[0]->[10]->[3] = 'A052405'; # decimal numbers without 3, OFFSET=0 start 0
$oeis_anum[1]->[10]->[4] = 'A052406'; # decimal numbers without 4, OFFSET=1 start 0
$oeis_anum[0]->[10]->[5] = 'A052413'; #                            OFFSET=0 start 0
$oeis_anum[0]->[10]->[6] = 'A052414'; #                            OFFSET=0
$oeis_anum[0]->[10]->[7] = 'A052419'; #                            OFFSET=0
$oeis_anum[0]->[10]->[8] = 'A052421'; #                            OFFSET=0
$oeis_anum[0]->[10]->[9] = 'A007095'; # "numbers in base 9"        OFFSET=0
# OEIS-Catalogue: A052382 radix=10 digit=0
# OEIS-Catalogue: A052383 radix=10 digit=1
# OEIS-Catalogue: A052404 radix=10 digit=2 i_start=0
# OEIS-Catalogue: A052405 radix=10 digit=3 i_start=0
# OEIS-Catalogue: A052406 radix=10 digit=4
# OEIS-Catalogue: A052413 radix=10 digit=5 i_start=0
# OEIS-Catalogue: A052414 radix=10 digit=6 i_start=0
# OEIS-Catalogue: A052419 radix=10 digit=7 i_start=0
# OEIS-Catalogue: A052421 radix=10 digit=8 i_start=0
# OEIS-Catalogue: A007095 radix=10 digit=9 i_start=0  # base 10 no 9

sub oeis_anum {
  my ($self) = @_;
  my $radix = $self->{'radix'};
  my $digit = $self->{'digit'};
  if ($digit == -1) {
    $digit = $radix-1;
  }
  return $oeis_anum[$self->i_start]->[$radix]->[$digit];
}


#------------------------------------------------------------------------------

# sub rewind {
#   my ($self) = @_;
# 
#   my $radix = $self->{'radix'};
#   my $digit = $self->{'digit'};
#   if ($digit == -1) { $digit = $radix - 1; }
#   $digit = $digit % $radix;
# 
#   # my $lo = 0;
#   # my $n = 0;
#   # my $i = $self->i_start;
#   # if ($radix == 2) {
#   #   if ($digit == 1) {
#   #     my $n = 1;
#   #     while ($n < $lo) {
#   #       $i++;
#   #     }
#   #   }
#   # } else {
#   #   # look at the $radix digits of $n, build $i by treating as $radix-1,
#   #   # increment any $digit to go to the next without that
#   #   my $power = 1;
#   #   while ($n) {
#   #     my $rem = $n % $radix;
#   #     if ($rem >= $digit) {
#   #       $n++;
#   #     } else {
#   #       $i += $rem * $power;
#   #     }
#   #     $n = int ($n / $radix);
#   #     $power *= ($radix-1);
#   #   }
#   # 
#   #   if ($lo < 0) {
#   #     $i = -$i;
#   #     if ($n == $lo) {
#   #       $i--;
#   #     }
#   #   }
#   # }
# 
#   $self->Math::NumSeq::Base::IterateIth::rewind();
# }

# without 0
#     1-9 1-9 ... 1-9 for len digits
#     each is 9^len many
#     start at i = 9^1 + ... + 9^(len-1)
#                = 9^0 + 9^1 + ... + 9^(len-1)  - 1
#                = (9^len - 1)/8 - 1
#                = (9^len - 1 - 8)/8
#                = (9^len - 9)/8
#     8*i + 1 = 9^level
#
#     add sentinel 1*9^len, so
#     from = i - (9^len - 9)/8 + 9^len
#          = i + (- 9^len + 9)/8 + 9^len
#          = i + (- 9^len + 9 + 8*9^len)/8
#          = i + (7*9^len + 9) / 8
#
#     and which is then a high digit 2*9^len too big
#
sub ith {
  my ($self, $i) = @_;
  ### RadixWithoutDigit ith(): $i

  $i -= $self->i_start;    # i_start=0 or i_start=1 both begin at value=0

  if (_is_infinite($i)) {
    return $i;  # don't loop forever if $i is +infinity
  }

  my $radix = $self->{'radix'};
  my $digit = $self->{'digit'};

  if ($radix == 2) {
    if ($digit == 0) {
      my $base = ($i < 30 ? 2 : Math::NumSeq::_to_bigint(2));
      return ($base ** ($i+1)) - 1;
    } else {
      ### radix 2 without 1 no values ...
      return undef;
    }
  }

  if ($digit == -1) {
    $digit = $radix-1;
  }

  my $r1 = $radix - 1;
  # if ($i < $r1) {
  #   return $i + ($i >= $digit);
  # }
  my $r2 = $radix - 2;
  ### $radix
  ### $r1
  ### $r2

  my $value = 0;
  if ($digit == 0) {
    my $len = 1;
    while (($r1**$len - $r1) / $r2 <= $i) {
      $len++;
    }
    $len--;
    ### $len
    ### base: ($r1**$len - $r1) / $r2
    ### sentinel: $r1**$len
    ### adj add: $r1**$len - ($r1**$len - $r1) / $r2
    ### adj formula: (($r2-1)*$r1**$len + $r1) / $r2

    ### assert: $i >= ($r1 ** $len - $r1) / $r2
    ### assert: $i < ($r1 ** $len - $r1) / $r2 + $r1 ** $len

    $i += (($r2-1)*$r1**$len + $r1) / $r2;
    $value = -2 * $radix**$len;
    ### i remainder: $i
    ### $value

    ### assert: $i >= $r1 ** $len
    ### assert: $i < 2 * $r1 ** $len
  }

  # $i converted to radix-1 digits, built back up as radix
  my $power = 1;
  while ($i > 0) {
    my $d = $i % $r1;
    $i = int($i/$r1);
    ### $value
    ### $d
    ### $power
    if ($d >= $digit) {
      $d++;
      ### inc to d: $d
    }
    $value += $power * $d;
    $power *= $radix;
  }
  ### stop at i: $i
  $value += $power * $i;

  ### $value
  return $value;

  # my $digit = 1;
  # my $x = $i;
  # while ($x) {
  #   ### x mod $radix: $x%$radix
  #   if (($x % $radix) == $digit) {
  #     ### add: $digit
  #     $i += $digit;
  #     $x++;
  #   }
  #   $x = int($x/$radix);
  #   $digit *= $radix;
  # }
  # return (($self->{'i'} = $i),
  #         1);
}

sub pred {
  my ($self, $value) = @_;

  my $radix = $self->{'radix'};
  my $digit = $self->{'digit'};
  if ($digit == -1) {
    $digit = $radix-1;
  }

  if ($value < 0
      || ($digit == 0 && $value == 0)
      || $value != int($value)
      || _is_infinite($value)) {  # don't loop forever if $value is +infinity
    return 0;
  }

  while ($value) {
    my $got_digit = $value % $radix;
    if ($got_digit == $digit) {
      return 0; # contains $digit, so no
    }
    $value = int(($value-$got_digit) / $radix);  # subtract to stay in UV integer
  }
  return 1;
}

1;
__END__

=for stopwords Ryde Math-NumSeq

=head1 NAME

Math::NumSeq::RadixWithoutDigit -- integers without a given digit

=head1 SYNOPSIS

 use Math::NumSeq::RadixWithoutDigit;
 my $seq = Math::NumSeq::RadixWithoutDigit->new;
 my ($i, $value) = $seq->next;

=head1 DESCRIPTION

The integers without a given digit, for example decimal without 9s is 0 to
8, 10 to 18, 20 to 28, ... 80 to 88, 100, etc.

=head1 FUNCTIONS

See L<Math::NumSeq/FUNCTIONS> for behaviour common to all sequence classes.

=over 4

=item C<$seq = Math::NumSeq::RadixWithoutDigit-E<gt>new (radix =E<gt> $r, digit =E<gt> $d)>

Create and return a new sequence object.

=back

=head2 Random Access

=over

=item C<$value = $seq-E<gt>ith($i)>

Return the C<$i>'th number which doesn't have the digit.

=item C<$bool = $seq-E<gt>pred($value)>

Return true if C<$value> doesn't have the given C<digit>.

=back

=head1 SEE ALSO

L<Math::NumSeq>

=head1 HOME PAGE

L<http://user42.tuxfamily.org/math-numseq/index.html>

=head1 LICENSE

Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 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
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