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A002819
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Liouville's function L(n) = partial sums of A008836.
(Formerly M0042 N0012)
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17
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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, -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, -4, -5, -6, -7, -6, -7, -8, -7, -8, -9, -10, -9, -8, -9, -8, -7, -6
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| 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
Prime number theorem is equivalent to a(n)=o(n). - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 02 2003
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REFERENCES
| B. Cloitre, A tauberian approach to RH, Arxiv preprint arXiv:1107.0812, 2011
H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409.
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. T. Haimo, Experimentation and Conjecture Are Not Enough, The American Mathematical Monthly Volume 102 Number 2, 1995, page 105.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Tanaka, A numerical investigation on cumulative sum of the Liouville function, Tokyo J. Math. 3 (1980), 187-189.
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..10000
Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, Sign changes in sums of the Liouville function. Math. Comp. 77 (2008), 1681-1694.
Eric Weisstein's World of Mathematics, Liouville Function
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FORMULA
| a(n)=determinant of A174856. [From Mats Granvik (mats.granvik(AT)abo.fi), Mar 31 2010]
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MAPLE
| A002819 := n -> add((-1)^numtheory[bigomega](i), i=1..n): # Peter Luschny, Sep 15 2011
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MATHEMATICA
| Accumulate[Join[{0}, LiouvilleLambda[Range[90]]]] (* From Harvey P. Dale, Nov 08 2011 *)
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PROG
| (PARI) a(n)=sum(i=1, n, (-1)^bigomega(i))
(Haskell)
a002819 n = a002819_list !! n
a002819_list = scanl (+) 0 a008836_list
-- Reinhard Zumkeller, Nov 19 2011
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CROSSREFS
| Cf. A008836, A002053, A028488.
Sequence in context: A106509 A196199 A053615 * A037834 A179765 A004074
Adjacent sequences: A002816 A002817 A002818 * A002820 A002821 A002822
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KEYWORD
| nice,sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Jul 09 2001
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