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Search: id:A002819
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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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+0
8
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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; listen)
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OFFSET
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0,9
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COMMENT
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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
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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
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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. [From T. D. Noe (noe(AT)sspectra.com), Jul 17 2009]
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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PROGRAM
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(PARI) a(n)=sum(i=1, n, (-1)^bigomega(i))
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CROSSREFS
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Cf. A008836, A002053, A028488.
Sequence in context: A116433 A106509 A053615 this_sequence A037834 A004074 A053646
Adjacent sequences: A002816 A002817 A002818 this_sequence A002820 A002821 A002822
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KEYWORD
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nice,sign,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jul 09 2001
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