Example Möbius transformations (Sloane and Plouffe 1995, p. 22) include for all , giving the inverse transform as , 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ... (OEIS A000005),
the divisor function of . The Möbius transform of gives , 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, ... (OEIS A000010),
the totient function of . The inverse Möbius transform of the sequence and gives , 4, 0, 4, 8, 0, 0, 4, 4, ... (OEIS A004018),
the number of ways
of writing
as a sum of two squares. The inverse Möbius transform of for prime and for composite gives the sequence , 1, 1, 1, 1, 2, 1, 1, 1, ... (OEIS A001221),
the number of distinct prime factors of
.
Bender, E. A. and Goldman, J. R. "On the Applications of Möbius Inversion in Combinatorial Analysis." Amer. Math.
Monthly82, 789-803, 1975.Bernstein, M. and Sloane, N. J. A.
"Some Canonical Sequences of Integers." Linear Algebra Appl.226/228,
57-72, 1995.Gessel, I. and Rota, C.-G. (Eds.). Classic
Papers in Combinatorics. Boston, MA: Birkhäuser, 1987.Hardy,
G. H. and Wright, E. M. §17.10 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, 1979.Rota, G.-C. "On the Foundations of Combinatorial Theory
I. Theory of Möbius Functions." Z. für Wahrscheinlichkeitsth.2,
340-368, 1964.Sloane, N. J. A. Sequences A000005/M0246,
A000010/M0299, A001221/M0056,
and A004018/M3218 in "The On-Line Encyclopedia
of Integer Sequences."Sloane, N. J. A. and Plouffe, S.
The
Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.Stanley,
R. P. Enumerative
Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press,
p. 259, 1999.
, 
, ... with
,
, ... via the Möbius
inversion formula,
to 
is sometimes called the sum-of-divisors transform. Two other equivalent formulations
are given by
is the Riemann zeta function (Sloane and
Plouffe 1995, p. 21).
for all 
, giving the inverse transform as 
, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ... (OEIS A000005),
the divisor function 
of 
. The Möbius transform of 
gives 
, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, ... (OEIS A000010),
the totient function of 
. The inverse Möbius transform of the sequence 
and 
gives 
, 4, 0, 4, 8, 0, 0, 4, 4, ... (OEIS A004018),
the number of ways 
of writing 
as a sum of two squares. The inverse Möbius transform of 
for 
prime and 
for 
composite gives the sequence 
, 1, 1, 1, 1, 2, 1, 1, 1, ... (OEIS A001221),
the number of distinct prime factors of
.
