Mobius Function Calculator

This Mobius function calculator computes mu of n, a function at the heart of analytic number theory. The rule is simple: mu(n) is 0 if n is divisible by the square of any prime, otherwise it is plus 1 if n has an even number of distinct prime factors and minus 1 if it has an odd number, with mu(1) defined as 1. Enter a number up to a billion and the calculator returns mu(n), whether n is squarefree, and its count of distinct primes. The Mobius function powers Mobius inversion, links to the distribution of primes through the Riemann zeta function, and appears throughout combinatorics. The definition, a worked example and the assumptions are set out below.

Whole number n (1 to 1,000,000,000)

Squarefree?

Distinct prime factors

The formula

mu(1) = 1. If n has a squared prime factor, mu(n) = 0. Otherwise mu(n) = (minus 1) to the power k, where k is the number of distinct prime factors. So squarefree numbers give plus or minus 1, by parity.

Worked example

30 = 2 times 3 times 5, three distinct primes and squarefree, so mu(30) = (minus 1) cubed = minus 1. But 12 = 2 squared times 3 has a squared factor, so mu(12) = 0. Enter 30 to confirm.

Frequently asked questions

What does the Mobius function do?

It returns 0 for numbers with a squared prime factor, and plus or minus 1 for squarefree numbers depending on whether they have an even or odd number of prime factors.

What is a squarefree number?

One not divisible by any perfect square above 1, equivalently no prime appears twice in its factorisation.

Where is it used?

In Mobius inversion, in counting problems, and in deep links between primes and the Riemann zeta function.

Who this calculator is for

This calculator is for students, programmers and anyone exploring number theory.

What this calculator assumes

You enter a whole number within the stated range.
The exact integer algorithm is used.
Very large results are shown grouped for readability.

Formula and sources

Related calculators

Prime Factorization Calculator: break a number into prime factors.
LCM Calculator: lowest common multiple.
GCF Calculator: greatest common factor.

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