Search: id:A002819
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%I A002819 M0042 N0012
%S A002819 0,1,0,1,0,1,0,1,2,1,0,1,2,3,2,1,0,1,2,3,4,3,2,3,2,1,0,1,2,3,4,5,6,5,4,
%T A002819 3,2,3,2,1,0,1,2,3,4,5,4,5,6,5,6,5,6,7,6,5,4,3,2,3,2,3,2,3,2,1,2,3,4,3,
%U A002819 4,5,6,7,6,7,8,7,8,9,10,9,8,9,8,7,6,5,4,5,4,3,4,3
%V A002819 0,1,0,-1,0,-1,0,-1,-2,-1,0,-1,-2,-3,-2,-1,0,-1,-2,-3,-4,-3,-2,-3,-2,-1,
               0,-1,-2,-3,-4,
%W A002819 -5,-6,-5,-4,-3,-2,-3,-2,-1,0,-1,-2,-3,-4,-5,-4,-5,-6,-5,-6,-5,-6,-7,-6,
               -5,-4,-3,-2,
%X A002819 -3,-2,-3,-2,-3,-2,-1,-2,-3,-4,-3,-4,-5,-6,-7,-6,-7,-8,-7,-8,-9,-10,-9,
               -8,-9,-8,-7,-6
%N A002819 Liouville's function L(n) = partial sums of A008836.
%C A002819 Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper 
               Haimo. George Polya conjectured 1919 that L(n) <= 0 for all n >= 
               2. The conjecture was generally deemed true for nearly 40 years, 
               until 1958, when C. B. Haselgrove proved that L(n) > 0 for infinitely 
               many n. In 1962, R. S. Lehman found that L(906180359) = 1 and in 
               1980, M. Tanaka discovered that the smallest counterexample of the 
               Polya conjecture occurs when n = 906150257. - Harri Ristiniemi (harri.ristiniemi(AT)nicf.), 
               Jun 23 2001
%C A002819 Prime number theorem is equivalent to a(n)=o(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Feb 02 2003
%D A002819 H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy 
               of Sciences. Section A, 12 (1940), 407-409.
%D A002819 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 A002819 D. T. Haimo, Experimentation and Conjecture Are Not Enough, The American 
               Mathematical Monthly Volume 102 Number 2, 1995, page 105.
%D A002819 R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
%D A002819 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002819 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002819 M. Tanaka, A numerical investigation on cumulative sum of the Liouville 
               function, Tokyo J. Math. 3 (1980), 187-189.
%H A002819 T. D. Noe, <a href="b002819.txt">Table of n, a(n) for n=0..10000</a>
%H A002819 Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, <a href="http:/
               /www.ams.org/mcom/2008-77-263/S0025-5718-08-02036-X/home.html">Sign 
               changes in sums of the Liouville function. Math. Comp. 77 (2008), 
               1681-1694.</a> [From T. D. Noe (noe(AT)sspectra.com), Jul 17 2009]
%H A002819 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LiouvilleFunction.html">Link to a section of The World of Mathematics.</
               a>
%o A002819 (PARI) a(n)=sum(i=1,n,(-1)^bigomega(i))
%Y A002819 Cf. A008836, A002053, A028488.
%Y A002819 Sequence in context: A116433 A106509 A053615 this_sequence A037834 A004074 
               A053646
%Y A002819 Adjacent sequences: A002816 A002817 A002818 this_sequence A002820 A002821 
               A002822
%K A002819 nice,sign,easy
%O A002819 0,9
%A A002819 N. J. A. Sloane (njas(AT)research.att.com).
%E A002819 More terms from Larry Reeves (larryr(AT)acm.org), Jul 09 2001

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