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# 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`.