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

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Module Version: 72

# 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`.

# 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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