The Moebius(n) command computes the Moebius function of the positive integer n.

If n is divisible by the square of a prime number, then Moebius(n) is equal to 0. Otherwise, Moebius(n) is equal to 1 if n has an even number of prime factors, and is equal to -1 if n has an odd number of prime factors.

$\mathrm{M\öbius}$ and mu are aliases of Moebius.

You can enter the command mu using either the 1-D or 2-D calling sequence. For example, mu(8) is equivalent to $\mathrm{\mu }\left(8\right)$.

$\mathrm{with}\left(\mathrm{NumberTheory}\right)\:$

$\mathrm{Moebius}\left(1\right)$

$1$

$\mathrm{Moebius}\left(3^3\cdot 5\right)$

$0$

$\mathrm{Moebius}\left(3\cdot 5\cdot 7\right)$

$\mathrm{-1}$

$\mathrm{Moebius}\left(23\cdot 11\right)$

$1$

$\mathrm{igcd}\left(5657\,31945103\right)$

$1$

$\mathrm{\mu }\left(5657\cdot 31945103\right)$

$\mathrm{-1}$

$\mathrm{\mu }\left(5657\right)\mathrm{\mu }\left(31945103\right)$

$\mathrm{-1}$

$\mathrm{plots}:-\mathrm{pointplot}\left(\left[\mathrm{seq}\left(\left[n\,\mathrm{Moebius}\left(n\right)\right]\,n=1..50\right)\right]\,\mathrm{labels}=\left[''n''\,\mu \left(n\right)\right]\,\mathrm{symbol}=\mathrm{soliddiamond}\,\mathrm{symbolsize}=15\,\mathrm{color}=''OrangeRed''\,\mathrm{size}=\left[600\,400\right]\,\mathrm{tickmarks}=\left[\mathrm{default}\,\left[-1\,0\,1\right]\right]\right)$

The NumberTheory[Moebius] command was introduced in Maple 2016.

For more information on Maple 2016 changes, see Updates in Maple 2016.