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Möbius function calculator μ(n)

The Möbius function μ(n) is 0 if n has a squared prime factor, otherwise +1 or −1 for an even or odd number of distinct prime factors. Enter n and get μ(n) with its prime factorisation and whether n is squarefree.

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Enter Positive integer n and the Möbius function calculator μ(n) works out μ(n), Prime factorisation, Squarefree? straight away. For instance, with Positive integer n = 30 it returns μ(n) = -1, Prime factorisation = 2 × 3 × 5 and Squarefree? = yes.

How to use it

  1. Enter your values: Positive integer n.
  2. Read the result instantly: μ(n), Prime factorisation, Squarefree?.

Frequently asked questions

What does the Möbius function calculator μ(n) actually compute?

It takes Positive integer n and derives μ(n), Prime factorisation and Squarefree? from them. The calculation is live as you type, so the result updates on every change.

What information do I need to provide?

A single value: Positive integer n. Nothing else is required — no account, no file upload.

Can you show a worked example?

With Positive integer n = 30, the calculator returns μ(n) = -1, Prime factorisation = 2 × 3 × 5 and Squarefree? = yes. Those figures come from running this exact tool, so you can reproduce them by entering the same values.

What happens if I enter larger values?

It moves a lot. Using Positive integer n = 60 instead, μ(n) goes from -1 to 0 — which is why it is worth testing a few scenarios rather than trusting a single figure.

What does it give for smaller values?

Scaled down to Positive integer n = 15, μ(n) comes out at 1. The relationship is worth checking at both ends before you rely on a single result.

When would I actually use this?

Getting through a problem set: solving for the unknown, factoring an expression, and — more usefully — seeing the steps that got there.

What is the most common mistake?

Losing a solution when both sides are squared or divided by an expression. Squaring can add roots that do not satisfy the original equation, and dividing can remove one — check every answer back in the original.

What is the difference between the Möbius function calculator μ(n) and the Gamma function calculator Γ(x)?

This one returns μ(n) and Prime factorisation; the Gamma function calculator Γ(x) returns Γ(x) and (x−1)! for integer x. That is the whole difference — open the one whose figure you need.

Is there a tool for the next step?

Beta Function Calculator is the closest one after this: Compute the Euler Beta function B(x, y) = Γ(x)·Γ(y) / Γ(x+y) via a stable log-gamma.

What else is worth having open alongside it?

Complementary Error Function Calculator and Error Function Calculator — they come up in the same task often enough to be worth a second tab.

Further reading

All guides →
How-toHow to Use the Quadratic Formula: Solving ax² + bx + c = 0Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.How-toHow to Find a Cube Root by HandA step-by-step guide to finding a cube root by estimation and refinement — plus how it differs from a square root and when a value stays irrational.ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.