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.\"
.IX Title "Math::PlanePath::CoprimeColumns 3pm"
.TH Math::PlanePath::CoprimeColumns 3pm "2016-05-03" "perl v5.22.2" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
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.SH "NAME"
Math::PlanePath::CoprimeColumns \-\- coprime X,Y by columns
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::CoprimeColumns;
\& my $path = Math::PlanePath::CoprimeColumns\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "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.
.PP
.Vb 10
\&    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
.Ve
.PP
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.
.PP
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
.PP
.Vb 3
\&                          i=K
\&    cumulative totient = sum   phi(i)
\&                          i=1
.Ve
.PP
Anything making a straight line etc in the path will probably be related to
totient sums in some way.
.PP
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.
.SS "Direction Down"
.IX Subsection "Direction Down"
Option \f(CW\*(C`direction => \*(Aqdown\*(Aq\*(C'\fR reverses the order within each column to
go downwards to the X axis.
.PP
.Vb 1
\&    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
.Ve
.SS "N Start"
.IX Subsection "N Start"
The default is to number points starting N=0 as shown above.  An optional
\&\f(CW\*(C`n_start\*(C'\fR can give a different start with the same shape,  For example
to start at 1,
.PP
.Vb 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
.Ve
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes.
.ie n .IP """$path = Math::PlanePath::CoprimeColumns\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::CoprimeColumns\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::CoprimeColumns->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::CoprimeColumns\->new (direction => $str, n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::CoprimeColumns\->new (direction => $str, n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::CoprimeColumns->new (direction => $str, n_start => $n)"
.PD
Create and return a new path object.  \f(CW\*(C`direction\*(C'\fR (a string) can be
.Sp
.Vb 2
\&    "up"       (the default)
\&    "down"
.Ve
.ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4
.el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4
.IX Item "($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number \f(CW$n\fR on the path.  Points begin
at 0 and if \f(CW\*(C`$n < 0\*(C'\fR then the return is an empty list.
.ie n .IP """$bool = $path\->xy_is_visited ($x,$y)""" 4
.el .IP "\f(CW$bool = $path\->xy_is_visited ($x,$y)\fR" 4
.IX Item "$bool = $path->xy_is_visited ($x,$y)"
Return true if \f(CW\*(C`$x,$y\*(C'\fR is visited.  This means \f(CW$x\fR and \f(CW$y\fR have no
common factor.  This is tested with a \s-1GCD\s0 and is much faster than the full
\&\f(CW\*(C`xy_to_n()\*(C'\fR.
.SH "BUGS"
.IX Header "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.
.SH "OEIS"
.IX Header "OEIS"
This pattern is in Sloane's Online Encyclopedia of Integer Sequences in a
couple of forms,
.Sp
.RS 4
<http://oeis.org/A002088> (etc)
.RE
.PP
.Vb 4
\&    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
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::DiagonalRationals,
Math::PlanePath::RationalsTree,
Math::PlanePath::PythagoreanTree,
Math::PlanePath::DivisibleColumns
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
.PP
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the \s-1GNU\s0 General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
.PP
Math-PlanePath is distributed in the hope that it will be useful, but
\&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0
or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for
more details.
.PP
You should have received a copy of the \s-1GNU\s0 General Public License along with
Math-PlanePath.  If not, see <http://www.gnu.org/licenses/>.