Moebius Transform

Author

Eric W. Weisstein
May 8, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/MoebiusTransform.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MoebiusTransform.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Examples

Number of divisors

MoebiusTransform[Table[DivisorSigma[0,n],{n,10}]]

{InverseMoebiusTransform[Table[1,{20}]],
Table[DivisorSigma[0,n],{n,20}]}//ColumnForm

Totient function

{MoebiusTransform[Range[20]],
Table[EulerPhi[n],{n,20}]}//ColumnForm

Sum of two squares

b[n_Integer?EvenQ]:=0
b[n_Integer?OddQ]:=4(-1)^((n-1)/2)

{InverseMoebiusTransform[Table[b[n],{n,20}]],
Table[SquaresR[2,n],{n,20}]}//ColumnForm

Number of distinct primes dividing n (M0056)

InverseMoebiusTransform[Table[p[n],{n,50}]]