1Math::PlanePath::CoprimUesCeorluCmonnst(r3i)buted Perl DMoactuhm:e:nPtlaatnieoPnath::CoprimeColumns(3)
2
6 Math::PlanePath::CoprimeColumns -- coprime X,Y by columns
7
9 use Math::PlanePath::CoprimeColumns;
10 my $path = Math::PlanePath::CoprimeColumns->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path visits points X,Y which are coprime, ie. no common factor so
15 gcd(X,Y)=1, in columns from Y=0 to Y<=X.
16 13 | 63
18 12 | 57
19 11 | 45 56 62
20 10 | 41 55
21 9 | 31 40 54 61
22 8 | 27 39 53
23 7 | 21 26 30 38 44 52
24 6 | 17 37 51
25 5 | 11 16 20 25 36 43 50 60
26 4 | 9 15 24 35 49
27 3 | 5 8 14 19 29 34 48 59
28 2 | 3 7 13 23 33 47
29 1 | 0 1 2 4 6 10 12 18 22 28 32 42 46 58
30 Y=0|
31 +---------------------------------------------
32 X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
33 Since gcd(X,0)=0 the X axis itself is never visited, and since
35 gcd(K,K)=K the leading diagonal X=Y is not visited except X=1,Y=1.
36 The number of coprime pairs in each column is Euler's totient function
38 phi(X). Starting N=0 at X=1,Y=1 means N=0,1,2,4,6,10,etc horizontally
39 along row Y=1 are the cumulative totients
40 i=K
42 cumulative totient = sum phi(i)
43 i=1
44 Anything making a straight line etc in the path will probably be
46 related to totient sums in some way.
47 The pattern of coprimes or not within a column is the same going up as
49 going down, since X,X-Y has the same coprimeness as X,Y. This means
50 coprimes occur in pairs from X=3 onwards. When X is even the middle
51 point Y=X/2 is not coprime since it has common factor 2 from X=4
52 onwards. So there's an even number of points in each column from X=2
53 onwards and those cumulative totient totals horizontally along X=1 are
54 therefore always even likewise.
55 Direction Down
57 Option "direction => 'down'" reverses the order within each column to
58 go downwards to the X axis.
59 direction => "down"
61 8 | 22
63 7 | 18 23 numbering
64 6 | 12 downwards
65 5 | 10 13 19 24 |
66 4 | 6 14 25 |
67 3 | 4 7 15 20 v
68 2 | 2 8 16 26
69 1 | 0 1 3 5 9 11 17 21 27
70 Y=0|
71 +-----------------------------
72 X=0 1 2 3 4 5 6 7 8 9
73 N Start
75 The default is to number points starting N=0 as shown above. An
76 optional "n_start" can give a different start with the same shape, For
77 example to start at 1,
78 n_start => 1
80 8 | 28
82 7 | 22 27
83 6 | 18
84 5 | 12 17 21 26
85 4 | 10 16 25
86 3 | 6 9 15 20
87 2 | 4 8 14 24
88 1 | 1 2 3 5 7 11 13 19 23
89 Y=0|
90 +------------------------------
91 X=0 1 2 3 4 5 6 7 8 9
92
94 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
95 classes.
96 "$path = Math::PlanePath::CoprimeColumns->new ()"
98 "$path = Math::PlanePath::CoprimeColumns->new (direction => $str,
99 n_start => $n)"
100 Create and return a new path object. "direction" (a string) can be
101 "up" (the default)
103 "down"
104 "($x,$y) = $path->n_to_xy ($n)"
106 Return the X,Y coordinates of point number $n on the path. Points
107 begin at 0 and if "$n < 0" then the return is an empty list.
108 "$bool = $path->xy_is_visited ($x,$y)"
110 Return true if "$x,$y" is visited. This means $x and $y have no
111 common factor. This is tested with a GCD and is much faster than
112 the full "xy_to_n()".
113
115 The current implementation is fairly slack and is slow on medium to
116 large N. A table of cumulative totients is built and retained up to
117 the highest X column number used.
118
120 This pattern is in Sloane's Online Encyclopedia of Integer Sequences in
121 a couple of forms,
122 <http://oeis.org/A002088> (etc)
124 n_start=0 (the default)
126 A038567 X coordinate, reduced fractions denominator
127 A020653 X-Y diff, fractions denominator by diagonals
128 skipping N=0 initial 1/1
129 A002088 N on X axis, cumulative totient
131 A127368 by columns Y coordinate if coprime, 0 if not
132 A054521 by columns 1 if coprime, 0 if not
133 A054427 permutation columns N -> RationalsTree SB N X/Y<1
135 A054428 inverse, SB X/Y<1 -> columns
136 A121998 Y of skipped X,Y among 2<=Y<=X, those not coprime
137 A179594 X column position of KxK square unvisited
138 n_start=1
140 A038566 Y coordinate, reduced fractions numerator
141 A002088 N on X=Y+1 diagonal, cumulative totient
143
145 Math::PlanePath, Math::PlanePath::DiagonalRationals,
146 Math::PlanePath::RationalsTree, Math::PlanePath::PythagoreanTree,
147 Math::PlanePath::DivisibleColumns
148
150 <http://user42.tuxfamily.org/math-planepath/index.html>
151
153 Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
154 Kevin Ryde
155 Math-PlanePath is free software; you can redistribute it and/or modify
157 it under the terms of the GNU General Public License as published by
158 the Free Software Foundation; either version 3, or (at your option) any
159 later version.
160 Math-PlanePath is distributed in the hope that it will be useful, but
162 WITHOUT ANY WARRANTY; without even the implied warranty of
163 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
164 General Public License for more details.
165 You should have received a copy of the GNU General Public License along
167 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
168perl v5.34.0 2021-07-22Math::PlanePath::CoprimeColumns(3)