SYNOPSIS
use Math::PlanePath::DivisibleColumns;
my $path = Math::PlanePath::DivisibleColumns>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path visits points X,Y where X is divisible by Y going by columns from Y=1 to Y<=X.
18  57 17  51 16  49 15  44 14  40 13  36 12  34 11  28 10  26 9  22 56 8  19 48 7  15 39 6  13 33 55 5  9 25 43 4  7 18 32 47 3  4 12 21 31 42 54 2  2 6 11 17 24 30 38 46 53 1  0 1 3 5 8 10 14 16 20 23 27 29 35 37 41 45 50 52 Y=0 + X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Starting N=0 at X=1,Y=1 means the values 1,3,5,8,etc horizontally on Y=1 are the sums
i=K sum numdivisors(i) i=1
The current implementation is fairly slack and is slow on medium to large N.
Proper Divisors
"divisor_type => 'proper'" gives only proper divisors of X, meaning that Y=X itself is excluded.
9  39 8  33 7  26 6  22 38 5  16 29 4  11 21 32 3  7 13 20 28 37 2  3 6 10 15 19 25 31 36 1  0 1 2 4 5 8 9 12 14 17 18 23 24 27 30 34 35 Y=0 + X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
The pattern is the same, but the X=Y line skipped. The high line going up is at Y=X/2, when X is even, that being the highest proper divisor.
N Start
The default is to number points starting N=0 as shown above. An optional "n_start" can give a different start with the same shape, For example to start at 1,
n_start => 1 9  23 8  20 7  16 6  14 5  10 4  8 19 3  5 13 22 2  3 7 12 18 1  1 2 4 6 9 11 15 17 21 Y=0 + X=0 1 2 3 4 5 6 7 8 9
FUNCTIONS
See ``FUNCTIONS'' in Math::PlanePath for behaviour common to all path classes. "$path = Math::PlanePath::DivisibleColumns>new ()"
 "$path = Math::PlanePath::DivisibleColumns>new (divisor_type => $str, n_start => $n)"

Create and return a new path object. "divisor_type" (a string) can be
"all" (the default) "proper"
 "($x,$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 wellknown sqrt formula, a sum from 1 to sqrt(X).
S = floor(sqrt(X)) / i=S \ numdivs cumulative = 2 *  sum floor(X/i)   S^2 \ i=1 /
This means the N range for 0 to X can be calculated without working out all each column count 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)
n_start=0 (the default) A006218 N on Y=1 row, cumulative count of divisors A077597 N on X=Y diagonal, cumulative count divisors  1 n_start=1 A061017 X coord, each n appears countdivisors(n) times A027750 Y coord, list divisors of successive k A056538 X/Y, divisors high to low divisor_type=proper (and default n_start=0) A027751 Y coord divisor_type=proper, divisors of successive n (extra initial 1) divisor_type=proper, n_start=2 A208460 XY, being X subtract each proper divisor
A208460 has ``offset'' 2, hence "n_start=2" to match that. The same with all divisors would simply insert an extra 0 for the difference at X=Y.
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin RydeMathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.