Math::N...unction(3)

💖 NAME 💖

Math::NumSeq::MobiusFunction — bestie, it's the Mobius function sequence ✨ (a user-contributed Perl thing, filed under Math::N...unction(3), iykyk)

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🦋 SYNOPSIS 🦋

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

okay so literally three lines and you're in 💅

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🌸 DESCRIPTION 🌸

It's the sequence of the Mobius function, and here's the glow-up:

1, -1, -1, 0, -1, 1, ...
starting i=1

Each value is one of three things:

• 1 — if i has an even number of distinct prime factors
• -1 — if i has an odd number of distinct prime factors
• 0 — if i has a repeated prime factor

The sequence starts from i=1 and, like, it's reckoned as having no prime factors — that's 0 factors — and 0 counts as even, so value 1. Then i=2 and i=3 are value -1 because they each have one prime factor (they're primes, periodt), and i=4 is value 0 because it's 2*2, aka a repeated prime 2 💀

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✨ FUNCTIONS ✨

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes — that's the shared lore 🫶

• $seq = Math::NumSeq::MobiusFunction->new () — create and return a new sequence object. Fresh object, new era 👑

🎀 Random Access 🎀

• $value = $seq->ith($i) — return the Mobius function of $i, being 1, 0 or -1 according to the prime factors of $i.

ngl this calculation requires factorizing $i, and in the current code small primes are checked then a hard limit of 2**32 is placed on $i — all in the interests of not going into a near-infinite loop 😭

• $bool = $seq->pred($value) — return true if $value occurs in the sequence, which means simply 1, 0 or -1. That's it, that's the whole vibe check.

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🌈 SEE ALSO 🌈

• Math::NumSeq
• Math::NumSeq::LiouvilleFunction
• Math::NumSeq::PrimeFactorCount
• "moebius" in Math::Prime::Util

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🧚 HOME PAGE 🧚

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

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👑 LICENSE 👑

Copyright 2010, 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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and that's the whole page bestie, no cap 💅 — brought to you by perl v5.42.0, dated 2025-07-25, living in the Math::N...unction(3) corner of the manual. slay 🌟