1,1

Moebius inversion: f(n) = Sum_{ d divides n } g(d) for all n <=> g(n) = Sum_{ d divides n } mu(d)*f(n/d) for all n.

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).

A008683 = A140579^(-1) * A140664 - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 20 2008

See last sentence of abstract of Coons and Borwein: We give a new proof of Fatou's theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely multiplicative function from N to {-1,1), the series sum_{n=1 to infinity) f(n)z^n is transcendental over {Z}[z]; in particular, sum_{n=1 to infinity) lambda(n)z^n is transcendental, where lambda is Liouville's function. The transcendence of sum_{n=1 to infinity) mu(n)z^n is also proved. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 11 2008

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 826.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 24.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 262 and 287.

N. J. A. Sloane, First 10000 values of Mobius function: Table of n, a(n) for n = 1..10000

Joerg Arndt, Fxtbook

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, December 1972, p. 826.

G. J. Chaitin, [math/0306042] Thoughts on the Riemann hypothesis

K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)

Primefan, Mobius and Mertens Values For n=1 to 2500

R. P. Stanley, A combinatorial miscellany

G. Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Moebius function

Wolfram Research, First 50 values of mu(n)

Index entries for "core" sequences

Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, June 10, 2008.

mu(1)=1; mu(n)=(-1)^k if n is the product of k different primes; otherwise mu(n)=0.

Sum_{ d divides n } mu(d) = 1 if n=1 else 0.

Dirichlet generating function: Sum_{n >= 1} mu(n)/n^s = 1/zeta(s). Also Sum_{n >= 1} mu(n)*x^n/(1-x^n) = x.

phi(n) = Sum_{ d divides n } mu(d)*n/d.

Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1. - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.

a(n)=sum(d divides n, 2^A001221(d)*a(n/d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 05 2002

SUM_{d|n}(-1)^(n/d)*mobius(d) = 0. - Emeric Deutsch, Jan 28 2005

a(n) = (-1)^omega(n) * 0^(bigomega(n)-omega(n)) for n>0, where bigomega(n) and omega(n) are the numbers of prime factors of n with and without repetition (A001222, A001221, A046660). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Apr 05 2003

Dirichlet generating function for the absolute value: zeta(s)/zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005.

mu(n) = A129360 * (1, -1, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2007

with(numtheory): A008683 := n->mobius(n);

with(numtheory): [ seq(mobius(n), n=1..100) ];

Note that Maple defines mobius(0) to be -1. This is unwise! Moebius(0) is better left undefined.

Array[ MoebiusMu[ # ]&, 100, 0 ]

(AXIOM) [moebiusMu(n) for n in 1..100]

(MAGMA) [ MoebiusMu(n) : n in [1..100]];

(PARI) a(n)=moebius(n)

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1-X)[n])

Cf. A000010, A001221, A008966, A007423, A080847, A002321 (partial sums), A069158, A055615.

a(n) = A091219(A091202(n)).

Cf. A129360.

Cf. A140579, A140664.

Cf. A140254.

Sequence in context: A106510 A075437 A130047 this_sequence A008966 A080323 A069158

Adjacent sequences: A008680 A008681 A008682 this_sequence A008684 A008685 A008686

core,sign,easy,mult,nice

njas