Kevin Ryde > Math-NumSeq-69 > Math::NumSeq::CollatzSteps

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Module Version: 69   Source   Latest Release: Math-NumSeq-71

NAME ^

Math::NumSeq::CollatzSteps -- steps in the "3n+1" problem

SYNOPSIS ^

 use Math::NumSeq::CollatzSteps;
 my $seq = Math::NumSeq::CollatzSteps->new;
 my ($i, $value) = $seq->next;

DESCRIPTION ^

The number of steps it takes to reach 1 by the Collatz "3n+1" problem,

    0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, ...
    starting i=1

The Collatz problem iterates

    n -> / 3n+1  if n odd
         \ n/2   if n even

For example i=6 takes value=8 many steps to reach 1,

    6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

It's conjectured that any starting n will always eventually reduce to 1 and so the number of steps is finite. There's no limit in the code on how many steps counted. Math::BigInt is used if 3n+1 steps go past the usual scalar integer limit.

Up Steps

Option step_type => "up" counts only the 3n+1 up steps.

    step_type => "up"
    0, 0, 2, 0, 1, 2, 5, 0, 6, 1, 4, 2, 2, 5, 5, 0, 3, 6, 6, 1, 1, ...

This can also be thought of as steps iterating

    n -> (3n+1)/2^k  for maximum k

Down Steps

Option step_type => "down" counts only the n/2 down steps.

    step_type => "down"
    0, 1, 5, 2, 4, 6, 11, 3, 13, 5, 10, 7, 7, 12, 12, 4, 9, 14, ...

The total up+down gives the default "step_type=both".

Odd Numbers

Option on_values => "odd" counts steps on the odd numbers 2*i+1.

    on_values => "odd"
    0, 7, 5, 16, 19, 14, 9, 17, 12, 20, 7, 15, 23, 111, 18, 106, ...
    starting i=0 for odd number 1

Even Numbers

Option on_values => "even" counts steps on the even number 2*i,

    on_values => "even"
    1, 2, 8, 3, 6, 9, 17, 4, 20, 7, 15, 10, 10, 18, 18, 5, 13, 21, ...
    starting i=0 for even number 2

Since 2*i is even the first step is down n/2 to give i and thereafter the same as the plain count. This means the steps for "even" is simply 1 more than for plain "all".

FUNCTIONS ^

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::CollatzSteps->new ()
$seq = Math::NumSeq::CollatzSteps->new (step_type => $str, on_values => $str)

Create and return a new sequence object. The optional step_type parameter (a string) can be

    "up"      upward steps 3n+1
    "down"    downward steps n/2
    "both"    both up and down (the default)

The optional on_values parameter (a string) can be

    "all"     all integers i
    "odd"     odd integers 2*i+1
    "even"    even integers 2*i

Random Access

$value = $seq->ith($i)

Return the number of steps to take $i down to 1.

$bool = $seq->pred($value)

Return true if $value occurs as a step count. This is simply $value >= 0.

SEE ALSO ^

Math::NumSeq, Math::NumSeq::JugglerSteps, Math::NumSeq::ReverseAddSteps Math::NumSeq::HappySteps

HOME PAGE ^

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE ^

Copyright 2011, 2012, 2013, 2014 Kevin Ryde

Math-NumSeq 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-NumSeq 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-NumSeq. If not, see <http://www.gnu.org/licenses/>.

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