NAME

Math::PlanePath::DivisibleColumns -- X divisible by Y in columns

SYNOPSIS

use Math::PlanePath::DivisibleColumns;
my $path = Math::PlanePath::DivisibleColumns->new;
my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION

This path visits points X,Y where X is divisible by Y going by columns from Y=1 to Y<=X.

18 |                                                      57
17 |                                                   51
16 |                                                49
15 |                                             44
14 |                                          40
13 |                                       36
12 |                                    34
11 |                                 28
10 |                              26
 9 |                           22                         56
 8 |                        19                      48
 7 |                     15                   39
 6 |                  13                33                55
 5 |                9             25             43
 4 |             7          18          32          47
 3 |          4       12       21       31       42       54
 2 |       2     6    11    17    24    30    38    46    53
 1 |    0  1  3  5  8 10 14 16 20 23 27 29 35 37 41 45 50 52
Y=0|
   +---------------------------------------------------------
   X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18

Starting N=0 at X=1,Y=1 means the values 1,3,5,8,etc horizontally on Y=1 are the sums

 i=K
sum   numdivisors(i)
 i=1

The current implementation is fairly slack and is slow on medium to large N.

Proper Divisors

divisor_type => 'proper' gives only proper divisors of X, meaning that Y=X itself is excluded.

 9 |                                                      39   
 8 |                                                33         
 7 |                                          26               
 6 |                                    22                38   
 5 |                              16             29            
 4 |                        11          21          32         
 3 |                   7       13       20       28       37   
 2 |             3     6    10    15    19    25    31    36   
 1 |       0  1  2  4  5  8  9 12 14 17 18 23 24 27 30 34 35
Y=0|
   +---------------------------------------------------------
   X=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18

The pattern is the same, but the X=Y line skipped. The high line going up is at Y=X/2, when X is even, that being the highest proper divisor.

FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.

$path = Math::PlanePath::DivisibleColumns->new ()

$path = Math::PlanePath::DivisibleColumns->new (divisor_type => 'proper')

Create and return a new path object.

($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path. Points begin at 0 and if $n < 0 then the return is an empty list.

SEE ALSO

Math::PlanePath, Math::PlanePath::CoprimeColumns

HOME PAGE

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE

Copyright 2011 Kevin Ryde

Math-PlanePath 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-PlanePath 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-PlanePath. If not, see <http://www.gnu.org/licenses/>.