NAME
Math::PlanePath::CoprimeColumns  coprime X,Y by columns
SYNOPSIS
use Math::PlanePath::CoprimeColumns;
my $path = Math::PlanePath::CoprimeColumns>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path visits points X,Y which are coprime, ie. no common factor so gcd(X,Y)=1, in columns from Y=0 to Y<=X.
13  63
12  57
11  45 56 62
10  41 55
9  31 40 54 61
8  27 39 53
7  21 26 30 38 44 52
6  17 37 51
5  11 16 20 25 36 43 50 60
4  9 15 24 35 49
3  5 8 14 19 29 34 48 59
2  3 7 13 23 33 47
1  0 1 2 4 6 10 12 18 22 28 32 42 46 58
Y=0
+
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Since gcd(X,0)=0 the X axis itself is never visited, and since gcd(K,K)=K the leading diagonal X=Y is not visited except X=1,Y=1.
The number of coprime pairs in each column is Euler's totient function phi(X). Starting N=0 at X=1,Y=1 means N=0,1,2,4,6,10,etc horizontally along row Y=1 are the cumulative totients
i=K
cumulative totient = sum phi(i)
i=1
Anything making a straight line etc in the path will probably be related to totient sums in some way.
The pattern of coprimes or not within a column is the same going up as going down, since X,XY has the same coprimeness as X,Y. This means coprimes occur in pairs from X=3 onwards. When X is even the middle point Y=X/2 is not coprime since it has common factor 2 from X=4 onwards. So there's an even number of points in each column from X=2 onwards and those cumulative totient totals horizontally along X=1 are therefore always even likewise.
Direction Down
Option direction => 'down'
reverses the order within each column to go downwards to the X axis.
direction => "down"
8  22
7  18 23 numbering
6  12 downwards
5  10 13 19 24 
4  6 14 25 
3  4 7 15 20 v
2  2 8 16 26
1  0 1 3 5 9 11 17 21 27
Y=0
+
X=0 1 2 3 4 5 6 7 8 9
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
8  28
7  22 27
6  18
5  12 17 21 26
4  10 16 25
3  6 9 15 20
2  4 8 14 24
1  1 2 3 5 7 11 13 19 23
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::CoprimeColumns>new ()
$path = Math::PlanePath::CoprimeColumns>new (direction => $str, n_start => $n)

Create and return a new path object.
direction
(a string) can be"up" (the default) "down"
($x,$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. $bool = $path>xy_is_visited ($x,$y)

Return true if
$x,$y
is visited. This means$x
and$y
have no common factor. This is tested with a GCD and is much faster than the fullxy_to_n()
.
BUGS
The current implementation is fairly slack and is slow on medium to large N. A table of cumulative totients is built and retained up to the highest X column number used.
OEIS
This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a couple of forms,
http://oeis.org/A002088 (etc)
n_start=0 (the default)
A038567 X coordinate, reduced fractions denominator
A020653 XY diff, fractions denominator by diagonals
skipping N=0 initial 1/1
A002088 N on X axis, cumulative totient
A127368 by columns Y coordinate if coprime, 0 if not
A054521 by columns 1 if coprime, 0 if not
A054427 permutation columns N > RationalsTree SB N X/Y<1
A054428 inverse, SB X/Y<1 > columns
A121998 Y of skipped X,Y among 2<=Y<=X, those not coprime
A179594 X column position of KxK square unvisited
n_start=1
A038566 Y coordinate, reduced fractions numerator
A002088 N on X=Y+1 diagonal, cumulative totient
SEE ALSO
Math::PlanePath, Math::PlanePath::DiagonalRationals, Math::PlanePath::RationalsTree, Math::PlanePath::PythagoreanTree, Math::PlanePath::DivisibleColumns
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
MathPlanePath 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/>.