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A002819 Liouville's function L(n) = partial sums of A008836.
(Formerly M0042 N0012)
25
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; text; internal format)
OFFSET

0,9

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, Feb 02 2003

All integers appear infinitely often in this sequence. - Charles R Greathouse IV, Aug 20 2016

REFERENCES

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.

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).

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.

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. [Annotated scanned copy]

H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]

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.

M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3, 187-189, 1980.

Eric Weisstein's World of Mathematics, Liouville Function

FORMULA

a(n) = determinant of A174856. - Mats Granvik, Mar 31 2010

MAPLE

A002819 := n -> add((-1)^numtheory[bigomega](i), i=1..n): # Peter Luschny, Sep 15 2011

MATHEMATICA

Accumulate[Join[{0}, LiouvilleLambda[Range[90]]]] (* Harvey P. Dale, Nov 08 2011 *)

PROG

(PARI) a(n)=sum(i=1, n, (-1)^bigomega(i))

(PARI) a(n)=my(v=vectorsmall(n, i, 1)); forprime(p=2, sqrtint(n), for(e=2, logint(n, p), forstep(i=p^e, n, p^e, v[i]*=-1))); forprime(p=2, n, forstep(i=p, n, p, v[i]*=-1)); sum(i=1, #v, v[i]) \\ Charles R Greathouse IV, Aug 20 2016

(Haskell)

a002819 n = a002819_list !! n

a002819_list = scanl (+) 0 a008836_list

-- Reinhard Zumkeller, Nov 19 2011

CROSSREFS

Cf. A008836, A002053, A028488, A239122.

Sequence in context: A255175 A196199 A053615 * A037834 A212496 A179765

Adjacent sequences:  A002816 A002817 A002818 * A002820 A002821 A002822

KEYWORD

nice,sign

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 09 2001

STATUS

approved

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Last modified May 29 07:29 EDT 2017. Contains 287243 sequences.