NAME
Math::PlanePath::MathImageDivisibleColumns -- x divisible by y in columns
SYNOPSIS
use Math::PlanePath::MathImageDivisibleColumns;
my $path = Math::PlanePath::MathImageDivisibleColumns->new;
my ($x, $y) = $path->n_to_xy (123);DESCRIPTION
This path visits points X,Y where X is divisible by Y, in columns from Y=1 to Y<=X.
18 | 57
17 | 51
16 | 49
15 | 44
14 | 40
13 | 36
12 | 34
11 | 28
10 | 26
9 | 22 56
8 | 19 48
7 | 15 39
6 | 13 33 55
5 | 9 25 43
4 | 7 18 32 47
3 | 4 12 21 31 42 54
2 | 2 6 11 17 24 30 38 46 53
1 | 0 1 3 5 8 10 14 16 20 23 27 29 35 37 41 45 50 52
Y=0|
+---------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18The number of divisors in each column is ..., and starting N=0 at X=1,Y=1 means the values 1,3,5,8,etc horizontally along X=1 are the sums
i=K
sum numdivisors(i)
i=1The pattern of divisors or not is the same going up a column as going down, since X,X-Y has the same coprimeness as X,Y. This means coprimes occur in pairs from X=3 onwards. (In X even the middle point Y=X/2 is not coprime since they have common factor 2, from X=4 onwards.) So there's an even number of points in each column from X=2 onwards and the totals horizontally along X=1 are even likewise.
The current implementation is pretty slack and is fairly slow on medium to large N, but the resulting pattern is interesting. Anything making a straight line etc in the path will probably have to be related to phi sums in some way.
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for the behaviour common to all path classes.
$path = Math::PlanePath::MathImageDivisibleColumns->new ()-
Create and return a new path object.
($x,$y) = $path->n_to_xy ($n)-
Return the X,Y coordinates of point number
$non the path. Points begin at 0 and if$n < 0then the return is an empty list.