%I A008683
%S A008683 1,1,1,0,1,1,1,0,0,1,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,
%T A008683 1,0,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,1,0,0,1,1,0,0,0,1,
%U A008683 0,1,0,1,0,1,1,1,0,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,1
%V A008683 1,-1,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0,-1,0,-1,0,1,1,-1,0,0,
%W A008683 1,0,0,-1,-1,-1,0,1,1,1,0,-1,1,1,0,-1,-1,-1,0,0,1,-1,0,0,0,1,
%X A008683 0,-1,0,1,0,1,1,-1,0,-1,1,0,0,1,-1,-1,0,1,-1,-1,0,-1,1,0,0,1
%N A008683 Moebius (or Mobius) function mu(n).
%C A008683 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.
%C A008683 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).
%C A008683 A008683 = A140579^(-1) * A140664 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 20 2008
%C A008683 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
%C A008683 Equals row sums of triangle A144735 (the square of triangle A054533). 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]
%C A008683 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
%C A008683 Contribution from Mats Granvik (mats.granvik(AT)abo.fi), Dec 06 2008: 
               (Start)
%C A008683 The Editorial Office of the Journal of Number Theory kindly provided
%C A008683 (via B. Conrey) the following proof of the conjecture:
%C A008683 Let A be A143104 and B be A143104 where T(n,n)=0.
%C A008683 "Suppose you expand det(B_n) along the bottom row. There is only a 1 
               in the
%C A008683 first position and so the answer is (-1)^n times det(C_{n-1}) say, where
%C A008683 C_{n-1} is the (n-1) by (n-1) matrix obtained from B_n by deleting the 
               first
%C A008683 column and the last row.
%C A008683 Now the determinant of the Redheffer matrix is det(A_n)=M(n) where M(n) 
               is
%C A008683 the sum of mu(m) for 1<=m<=n. Expanding det(A_n) along the bottom row, 
               we see that
%C A008683 det(A_n)=(-1)^n*det(C_{n-1})+M(n-1). So we have
%C A008683 det(B_n)=(-1)^n*det(C_{n-1})=det(A_n)-M(n-1)=M(n)-M(n-1)=mu(n)."
%C A008683 (End)
%C A008683 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
%C A008683 Convolved with A152902 = A000027, the natural numbers. [From Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Dec 14 2008]
%C A008683 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 13 2009: 
               (Start)
%C A008683 [Pickover, p. 226]: "The probability that a number falls in the -1 mailbox
%C A008683 turns out to be 3/Pi^2 - the same probability as for falling in the +1 
               mailbox". (End)
%D A008683 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, Tenth Printing, 
               1972, p. 826.
%D A008683 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 24.
%D A008683 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 161, #16.
%D A008683 Deleglise, Marc and Rivat, Joel, Computing the summation of the Mobius 
               function. Experiment. Math. 5 (1996), no. 4, 291-295.
%D A008683 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, th. 262 and 287.
%D A008683 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]
%D A008683 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 226.
%H A008683 Daniel Forgues, <a href="b008683.txt">Table of n, a(n) for n=1..100000</
               a>
%H A008683 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Fxtbook</a>
%H A008683 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A008683 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 
               1972, p. 826.
%H A008683 G. J. Chaitin, <a href="http://arXiv.org/abs/math.HO/0306042">[math/0306042] 
               Thoughts on the Riemann hypothesis</a>
%H A008683 Michael Coons and Peter Borwein, <a href="http://arxiv.org/pdf/0806.1563">
               Transcendence of Power Series for Some Number Theoretic Functions</
               a>, June 10, 2008.
%H A008683 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing 
               n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H A008683 Ed Pegg Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_11_03_03.html">
               The Mobius function (and squarefree numbers)</a>
%H A008683 Primefan, <a href="http://primefan.tripod.com/Mertens2500.html">Mobius 
               and Mertens Values For n=1 to 2500</a>
%H A008683 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/comb.ps">
               A combinatorial miscellany</a>
%H A008683 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/
               TABLES/aaaFArit/MobiusMe.htm">Nombres de Moebius et de Mertens</a>
%H A008683 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MoebiusFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A008683 E. W. Weisstein, <a href="http://mathworld.wolfram.com/RedhefferMatrix.html">
               Redheffer Matrix</a>.
%H A008683 Wikipedia, <a href="http://en.wikipedia.org/wiki/Mobius_function">Moebius 
               function</a>
%H A008683 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/
               MoebiusMu/03/02">First 50 values of mu(n)</a>
%H A008683 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A008683 mu(1)=1; mu(n)=(-1)^k if n is the product of k different primes; otherwise 
               mu(n)=0.
%F A008683 Sum_{ d divides n } mu(d) = 1 if n=1 else 0.
%F A008683 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.
%F A008683 phi(n) = Sum_{ d divides n } mu(d)*n/d.
%F A008683 Multiplicative with a(p^e) = -1 if e = 1; 0 if e > 1. - David W. Wilson 
               (davidwwilson(AT)comcast.net), Aug 01, 2001.
%F A008683 a(n)=sum(d divides n, 2^A001221(d)*a(n/d)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 05 2002
%F A008683 SUM_{d|n}(-1)^(n/d)*mobius(d) = 0. - Emeric Deutsch, Jan 28 2005
%F A008683 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
%F A008683 Dirichlet generating function for the absolute value: zeta(s)/zeta(2s). 
               - Franklin T. Adams-Watters, Sep 11 2005.
%F A008683 mu(n) = A129360 * (1, -1, 0, 0, 0,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Apr 17 2007
%F A008683 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]
%F A008683 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]
%p A008683 with(numtheory): A008683 := n->mobius(n);
%p A008683 with(numtheory): [ seq(mobius(n), n=1..100) ];
%p A008683 Note that Maple defines mobius(0) to be -1. This is unwise! Moebius(0) 
               is better left undefined.
%p A008683 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]
%t A008683 Array[ MoebiusMu[ # ]&, 100, 0 ]
%o A008683 (AXIOM) [moebiusMu(n) for n in 1..100]
%o A008683 (MAGMA) [ MoebiusMu(n) : n in [1..100]];
%o A008683 (PARI) a(n)=moebius(n)
%o A008683 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1-X)[n])
%Y A008683 Cf. A000010, A001221, A008966, A007423, A080847, A002321 (partial sums), 
               A069158, A055615.
%Y A008683 a(n) = A091219(A091202(n)).
%Y A008683 Cf. A129360, A140579, A140664, A140254, A143104.
%Y A008683 A144735, A054533 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 
               2008]
%Y A008683 A152902 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 14 2008]
%Y A008683 Sequence in context: A075437 A130047 A157657 this_sequence A008966 A080323 
               A069158
%Y A008683 Adjacent sequences: A008680 A008681 A008682 this_sequence A008684 A008685 
               A008686
%K A008683 core,sign,easy,mult,nice
%O A008683 1,1
%A A008683 N. J. A. Sloane (njas(AT)research.att.com).
%E A008683 I changed the title of the Pickover reference Robert G. Wilson v (rgwv(AT)rgwv.com), 
               Aug 24 2009
%E A008683 Replaced a geocities.com URL - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 30 2009