Tool to calculate the value of the function μ (Mu) of Möbius (or Moebius) which has a value of -1, 0 or 1 according to its prime numbers decomposition.
Möbius Function - dCode
Tag(s) : Arithmetics
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Möbius Function
Mobius μ(N) Calculator
Answers to Questions (FAQ)
What is the Mobius Mu function? (Definition)
The function $ μ(n) $, called the Möbius function (or Moebius), is defined for any integer $ n > 0 $ of the set $ \mathbb{N}* $ in the set of 3 values $ \{ -1, 0, 1 \} $.
$ μ(n) $ is $ 0 $ if $ n $ has for divisor a perfect square (other than 1)
$ μ(n) $ is $ 1 $ if $ n $ has for divisors an even number of prime numbers
$ μ(n) $ is $ -1 $ if $ n $ has for divisors an odd number of prime numbers
How to calculate the value of the Mobius function?
Automatic method: indicate the value $ n $ for which to calculate $ μ(n) $ in dCode (above)
Manual method: the image of $ μ(n) $ depends on the prime number decomposition of $ n $. If a prime number appears several times in the decomposition, then $ μ(n) = 0 $, otherwise, if the decomposition has an even number of prime numbers, then $ μ(n) = 1 $ and otherwise with an odd number of prime numbers $ μ(n) = -1 $.
Example: $ 12 = 2 \times 2 \times 3 $ so $ μ(12) = 0 $ because $ 2 $ appears twice, and so $ 12 $ is divisible by $ 4 $, a perfect square
Example: $ 1234 = 2 \times 617 $ therefore $ μ(12) = 1 $ because the decomposition has 2 distinct primes (2 is an even number)
Example: $ 12345 = 3 \times 5 \times 823 $ so $ μ(12) = -1 $ because the decomposition has 3 distinct prime numbers (3 is an odd number)
Source code
dCode retains ownership of the "Möbius Function" source code. Any algorithm for the "Möbius Function" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Möbius Function" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) or any database download or API access for "Möbius Function" or any other element are not public (except explicit open source licence like Creative Commons). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Möbius Function" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source.
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
In a scientific article or book, the recommended bibliographic citation is: Möbius Function on dCode.fr [online website], retrieved on 2025-04-16,
