Calculate mu(n) from square-free status and prime factors
Result
Answer
Step-by-Step Derivation
Mobius Function Formula
mu(1) = 1
mu(n) = 0 if n is divisible by p^2
mu(n) = (-1)^k if n is square-free with k prime factors
The Mobius function depends on prime factorization. It first checks whether any prime factor appears more than once; if not, the sign is determined by the number of distinct prime factors.
⚠Enter a positive integer. The result is exact for normal browser-size integers, but very large values may need specialized factorization.
What Is the Mobius Function?
The Mobius function is a compact way to encode square-free structure. It is important because it reverses certain divisor sums through Mobius inversion.
Square-Free Check
Any repeated prime factor makes mu(n) equal 0.
Even Factor Count
Square-free n with even k gives mu(n)=1.
Odd Factor Count
Square-free n with odd k gives mu(n)=-1.
Inversion Use
The function is central in reversing divisor-sum formulas.
Example: For n = 30, the factorization is 2 x 3 x 5. It is square-free with 3 prime factors, so mu(30) = -1.
Applications of Mobius Function
Mobius InversionSquare-Free NumbersDivisor SumsMultiplicative FunctionsAnalytic Number Theory
Frequently Asked Questions
What is a Mobius function calculator?▼
It computes mu(n), a number theory function based on whether n is square-free and how many distinct prime factors n has.
What is the Mobius function formula?▼
mu(n)=0 if n is divisible by a square, mu(n)=1 for square-free n with an even number of prime factors, and -1 for an odd number.
How do I use this mu(n) calculator?▼
Enter a positive integer. The tool factors n, checks square-free status, and returns the Mobius value.
What does mu(n)=0 mean?▼
It means n is divisible by p^2 for at least one prime p, so n is not square-free.
Where is the Mobius function used?▼
It is used in Mobius inversion, multiplicative functions, divisor sums, analytic number theory, and combinatorics.
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