Mobius Function Calculator

This Mobius function calculator works out mu(n) for any whole number you enter, a value central to number theory that links directly to prime factorisation. Enter n as a whole number from 1 up to one billion, and the calculator returns mu(n) itself, whether n is squarefree, and the count of its distinct prime factors, so you can see exactly how the result was reached. The Mobius function follows a simple three-way rule: mu(1) equals 1; mu(n) equals 0 whenever n is divisible by the square of any prime; and for every other, squarefree, number, mu(n) is plus 1 if it has an even number of distinct prime factors and minus 1 if it has an odd number. Type in any number, whether a small textbook example or a large value with several prime factors, and the tool factorises it instantly rather than making you work through the arithmetic by hand. This makes it useful for students checking homework, programmers testing number-theory code, and anyone exploring how the Mobius function underpins Mobius inversion, counting arguments in combinatorics, and the deep connection between the primes and the Riemann zeta function. A worked example below shows how 30 and 12 give different results, one squarefree and one not, so you can confirm the calculator matches the definition before relying on it for your own numbers.

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

μ(30) = -1

Squarefree?Yes

Distinct prime factors3

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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