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 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.

FORMULAS

Rectangle to N Range

The cumulative divisor count up to and including a given X column can be calculated from the fairly well-known sqrt formula,

    S = floor(sqrt(X))
                           /   i=S             \
    divs cumulative = 2 * |   sum  floor(X/i)   |  - S^2
                           \   i=1             /

This means the N range for 0 to X can be calculated without needing to work out individual columns counts up to X. In the current code if column counts have been worked out then they're used, otherwise this formula.

OEIS

This pattern is in Sloane's Online Encyclopedia of Integer Sequences in the following forms,

    http://oeis.org/A061017  (etc)

    A061017    X coord, each n appears countdivisors(n) times
    A027750    Y coord, list divisors of successive n

    A027751    Y coord divisor_type=proper, divisors of successive n

    A006218    N on Y=1 row, cumulative count of divisors

SEE ALSO

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

HOME PAGE

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

LICENSE

Copyright 2011, 2012 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/>.

cpanm Math::PlanePath

perl -MCPAN -e shell
install Math::PlanePath