Provided by: libmath-planepath-perl_129-1_all 
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,X-Y 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 full "xy_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 X-Y 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/math-planepath/index.html>
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.