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

Equals row sums of triangle A144735 (the square of triangle A054533). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]

Conjecture: a(n) = determinant of Redheffer matrix A143104 where T(n,n)=0. Verified for 50 first terms. - Mats O. Granvik (mgranvik(AT)abo.fi), Jul 25 2008

Contribution from Mats Granvik (mats.granvik(AT)abo.fi), Dec 06 2008: (Start)

The Editorial Office of the Journal of Number Theory kindly provided

(via B. Conrey) the following proof of the conjecture:

Let A be A143104 and B be A143104 where T(n,n)=0.

"Suppose you expand det(B_n) along the bottom row. There is only a 1 in the

first position and so the answer is (-1)^n times det(C_{n-1}) say, where

C_{n-1} is the (n-1) by (n-1) matrix obtained from B_n by deleting the first

column and the last row.

Now the determinant of the Redheffer matrix is det(A_n)=M(n) where M(n) is

the sum of mu(m) for 1<=m<=n. Expanding det(A_n) along the bottom row, we see that

det(A_n)=(-1)^n*det(C_{n-1})+M(n-1). So we have

det(B_n)=(-1)^n*det(C_{n-1})=det(A_n)-M(n-1)=M(n)-M(n-1)=mu(n)."

(End)

Conjecture: Consider the table A051731 and treat 1 as a divisor. Move the value in the lower right corner vertically to a divisor position in the transpose of the table and you will find that the determinant is the Moebius function. The number of permutation matrices that contribute to the Moebius function appears to be A074206. - Mats Granvik (mats.granvik(AT)abo.fi), Dec 08 2008

Convolved with A152902 = A000027, the natural numbers. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 14 2008]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 13 2009: (Start)

[Pickover, p. 226]: "The probability that a number falls in the -1 mailbox

turns out to be 3/Pi^2 - the same probability as for falling in the +1 mailbox". (End)

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

Deleglise, Marc and Rivat, Joel, Computing the summation of the Mobius function. Experiment. Math. 5 (1996), no. 4, 291-295.

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

Clifford A. Pickover, "The Math Book, from Pythagoras to the 57-th Dimension, 250 Milestones in the History of Mathematics", Sterling Publishing, 2009, pg.226. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 13 2009]

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 226.

Daniel Forgues, Table of n, a(n) for n=1..100000

Joerg Arndt, Fxtbook

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

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

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

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

Ed Pegg Jr., The Mobius function (and squarefree numbers)

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.

E. W. Weisstein, Redheffer Matrix.

Wikipedia, Moebius function

Wolfram Research, First 50 values of mu(n)

Index entries for "core" sequences

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)gmail.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

mu(n) = -Sum_{d<n,d|n} mu(d) if n>1 and mu(1) = 1. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 13 2008]

It appears that: A008683(n+1) = determinant(mod(n;k)-mod(n+1;k)+1)/n!*(-1)^n [From Mats Granvik (mgranvik(AT)abo.fi), Aug 16 2008]

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.

with (numtheory): mu := proc(n::posint) option remember; if n=1 then 1 else -add (mu(d), d=divisors(n) minus{n}) fi end; seq (mu(n), n=1..77); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 13 2008]

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, A140579, A140664, A140254, A143104.

A144735, A054533 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]

A152902 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 14 2008]

Sequence in context: A075437 A130047 A157657 this_sequence A008966 A080323 A069158

Adjacent sequences: A008680 A008681 A008682 this_sequence A008684 A008685 A008686

core,sign,easy,mult,nice

N. J. A. Sloane (njas(AT)research.att.com).

I changed the title of the Pickover reference Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 24 2009

Replaced a geocities.com URL - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 30 2009