Math::N...tzSteps(3)

💖 Math::N...tzSteps(3) 💖

okay bestie, so this is the Math::NumSeq::CollatzSteps page, from the User Contributed Perl Documentation collection. it's giving math, it's giving chaos theory, it's giving "why is this so addictive". buckle up. 🎀

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✨ NAME ✨

Math::NumSeq::CollatzSteps — steps in the 3n+1 problem 🌀

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🌸 SYNOPSIS 🌸

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

literally three lines and you're in. periodt. 💅

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🦋 DESCRIPTION 🦋

so this sequence counts how many steps it takes to reach 1 using the Collatz 3n+1 problem. iconic behavior honestly.

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 like this:

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

for example, i=6 takes value=8 — that's 8 whole steps — to reach 1, like:

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

and the tea is: it's conjectured that any starting n will always eventually reduce to 1, so the number of steps is finite. no cap, nobody has proven it, we're all just vibing on faith here. 🫶

⚠️ there is no limit in the code on how many steps are counted, bestie. and Math::BigInt gets pulled in if the 3n+1 steps go past the usual scalar integer limit. so like, it won't break, it'll just get fancy. 💀

🔥 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, ...

you can also think of this 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, ...

and the total up+down is what gives you the default step_type=both. math is mathing. ✨

👑 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 after that it's the same as the plain count. which means the steps for "even" is simply 1 more than for plain "all". that's it, that's the whole trick. 🌈

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💜 FUNCTIONS 💜

peep "FUNCTIONS" in Math::NumSeq for the behaviour that's common to all sequence classes, bestie. this is just the local flavor.

• $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. yeah, that easy. 🙈

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🌸 SEE ALSO 🌸

• Math::NumSeq
• Math::NumSeq::JugglerSteps
• Math::NumSeq::ReverseAddSteps
• Math::NumSeq::HappySteps

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🎀 HOME PAGE 🎀

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

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💖 LICENSE 💖

Copyright 2011, 2012, 2013, 2014, 2016, 2019, 2020 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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okay that's the whole page, you ate it up. 💅 this was perl v5.42.0, dated 2025-07-25, and it lives at Math::N...tzSteps(3). go count some steps, bestie, ily 🫶✨