Kevin Ryde > Math-PlanePath > Math::PlanePath::CoprimeColumns

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Module Version: 117   Source  

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 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/>.

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