Search: id:A008836
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%I A008836
%S A008836 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A008836 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A008836 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%V A008836 1,-1,-1,1,-1,1,-1,-1,1,1,-1,-1,-1,1,1,1,-1,-1,-1,-1,1,1,-1,1,1,1,-1,-1,
               -1,-1,-1,-1,1,
%W A008836 1,1,1,-1,1,1,1,-1,-1,-1,-1,-1,1,-1,-1,1,-1,1,-1,-1,1,1,1,1,1,-1,1,-1,
               1,-1,1,1,-1,-1,
%X A008836 -1,1,-1,-1,-1,-1,1,-1,-1,1,-1,-1,-1,1,1,-1,1,1,1,1,1,-1,1,1,-1,1,1,1,
               1,-1,-1,-1,1,-1
%N A008836 Liouville's function lambda(n) = (-1)^k, where k is number of primes 
               dividing n (counted with multiplicity).
%C A008836 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 A008836 Denote by lambda(n) Liouville's function concerning the parity of the 
               number of prime divisors of n. Using a theorem of Allouche, Mendes 
               France and Peyriere and many classical results from the theory of 
               the distribution of prime numbers, we prove that lambda(n) is not 
               k-automatic for any k > 2. This yields that sum[n=1..infty] lambda(n) 
               X^n an element of F_p[[X]] is transcendental over F_p(X) for any 
               prime p > 2. Similar results are proved (or reproved) for many common 
               number--theoretic functions, including phi, mu, Omega, omega, rho 
               and others. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 
               22 2008]
%D A008836 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, page 37.
%D A008836 H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy 
               of Sciences. Section A, 12 (1940), 407-409.
%D A008836 H. Gupta, A table of values of Liouville's function L(n), Research Bulletin 
               of East Panjab University, No. 3 (Feb. 1950), 45-55.
%D A008836 R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
%D A008836 P. Ribenboim, Algebraic Numbers, p. 44.
%D A008836 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 279.
%H A008836 T. D. Noe, <a href="b008836.txt">Table of n, a(n) for n=1..10000</a>
%H A008836 Michael Coons and Peter Borwein, <a href="http://arxiv.org/pdf/0806.1563">
               Transcendence of Power Series for Some Number Theoretic Functions</
               a>
%H A008836 Michael Coons, <a href="http://arxiv.org/abs/0810.3709">(Non)Automaticity 
               of number theoretic functions</a>, Oct 21, 2008. [From Jonathan Vos 
               Post (jvospost3(AT)gmail.com), Oct 22 2008]
%H A008836 Weisstein, Eric W., <a href="http://mathworld.wolfram.com/LiouvilleFunction.html">
               Liouville Function</a>. [From Daniel Forgues (squid(AT)zensearch.com), 
               Mar 17 2009]
%F A008836 Dirichlet g.f.: zeta(2s)/zeta(s).
%F A008836 Sum_{ d divides n } lambda(d) = 1 if n is a square, else 0.
%F A008836 Completely multiplicative with a(p) = -1, p prime.
%F A008836 a(n) = (-1)^A001222(n) = (-1)^bigomega(n). - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Apr 16 2006
%F A008836 a(n) = A033999(A001222(n). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Sep 28 2009]
%p A008836 with(numtheory): A008836 := proc(n) local i,it,s; it := ifactors(n): 
               s := (-1)^add(it[2][i][2], i=1..nops(it[2])): RETURN(s) end:
%o A008836 (PARI) a(n)=if(n<1, 0, n=factor(n); (-1)^sum(i=1,matsize(n)[1],n[i,2]))
%Y A008836 Cf. A002053, A007421, A002819, A026424, A028260, A028488, A056912, A056913, 
               A001222, A065043, A066829.
%Y A008836 Cf. A001222.
%Y A008836 Sequence in context: A164660 A114523 A000012 this_sequence A064179 A106400 
               A112865
%Y A008836 Adjacent sequences: A008833 A008834 A008835 this_sequence A008837 A008838 
               A008839
%K A008836 sign,easy,nice,mult
%O A008836 1,1
%A A008836 N. J. A. Sloane (njas(AT)research.att.com).

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