Math::PlanePath::CoprimeColumns -- coprime X,Y by columns
use Math::PlanePath::CoprimeColumns; my $path = Math::PlanePath::CoprimeColumns->new; my ($x, $y) = $path->n_to_xy (123);
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.
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
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
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()
.
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.
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
Math::PlanePath, Math::PlanePath::DiagonalRationals, Math::PlanePath::RationalsTree, Math::PlanePath::PythagoreanTree, Math::PlanePath::DivisibleColumns
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2011, 2012, 2013, 2014, 2015, 2016 Kevin Ryde
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