++ed by:
BHANN EGOR

2 PAUSE users

Kevin Ryde

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.

FORMULAS

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/>.