NAME¶
Math::PlanePath::DivisibleColumns -- X divisible by Y in columns
SYNOPSIS¶
use Math::PlanePath::DivisibleColumns;
my $path = Math::PlanePath::DivisibleColumns->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION¶
This path visits points X,Y where X is divisible by Y going by columns from Y=1
to Y<=X.
18 | 57
17 | 51
16 | 49
15 | 44
14 | 40
13 | 36
12 | 34
11 | 28
10 | 26
9 | 22 56
8 | 19 48
7 | 15 39
6 | 13 33 55
5 | 9 25 43
4 | 7 18 32 47
3 | 4 12 21 31 42 54
2 | 2 6 11 17 24 30 38 46 53
1 | 0 1 3 5 8 10 14 16 20 23 27 29 35 37 41 45 50 52
Y=0|
+---------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Starting N=0 at X=1,Y=1 means the values 1,3,5,8,etc horizontally on Y=1 are the
sums
i=K
sum numdivisors(i)
i=1
The current implementation is fairly slack and is slow on medium to large N.
Proper Divisors¶
"divisor_type => 'proper'" gives only proper divisors of X, meaning
that Y=X itself is excluded.
9 | 39
8 | 33
7 | 26
6 | 22 38
5 | 16 29
4 | 11 21 32
3 | 7 13 20 28 37
2 | 3 6 10 15 19 25 31 36
1 | 0 1 2 4 5 8 9 12 14 17 18 23 24 27 30 34 35
Y=0|
+---------------------------------------------------------
X=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
The pattern is the same, but the X=Y line skipped. The high line going up is at
Y=X/2, when X is even, that being the highest proper divisor.
N Start¶
The default is to number points starting N=0 as shown above. An optional
"n_start" can give a different start with the same shape, For
example to start at 1,
n_start => 1
9 | 23
8 | 20
7 | 16
6 | 14
5 | 10
4 | 8 19
3 | 5 13 22
2 | 3 7 12 18
1 | 1 2 4 6 9 11 15 17 21
Y=0|
+------------------------------
X=0 1 2 3 4 5 6 7 8 9
FUNCTIONS¶
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
classes.
- "$path = Math::PlanePath::DivisibleColumns->new ()"
- "$path = Math::PlanePath::DivisibleColumns->new (divisor_type
=> $str, n_start => $n)"
- Create and return a new path object. "divisor_type" (a string)
can be
"all" (the default)
"proper"
- "($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.
Rectangle to N Range¶
The cumulative divisor count up to and including a given X column can be
calculated from the fairly well-known sqrt formula, a sum from 1 to sqrt(X).
S = floor(sqrt(X))
/ i=S \
numdivs cumulative = 2 * | sum floor(X/i) | - S^2
\ i=1 /
This means the N range for 0 to X can be calculated without working out all each
column count up to X. In the current code if column counts have been worked
out then they're used, otherwise this formula.
OEIS¶
This pattern is in Sloane's Online Encyclopedia of Integer Sequences in the
following forms,
n_start=0 (the default)
A006218 N on Y=1 row, cumulative count of divisors
A077597 N on X=Y diagonal, cumulative count divisors - 1
n_start=1
A061017 X coord, each n appears countdivisors(n) times
A027750 Y coord, list divisors of successive k
A056538 X/Y, divisors high to low
divisor_type=proper (and default n_start=0)
A027751 Y coord divisor_type=proper, divisors of successive n
(extra initial 1)
divisor_type=proper, n_start=2
A208460 X-Y, being X subtract each proper divisor
A208460 has "offset" 2, hence "n_start=2" to match that. The
same with all divisors would simply insert an extra 0 for the difference at
X=Y.
SEE ALSO¶
Math::PlanePath, Math::PlanePath::CoprimeColumns
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/>.