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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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31
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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;
text;
internal format)
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OFFSET
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0,9
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COMMENTS
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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, Feb 02 2003
In the Liouville function, every prime is assigned the value -1, so it may be expected that the values of a(n) are minimal (A360659) among all completely multiplicative sign functions. As it turns out, this is the case for n < 14 and n = 20. For any other n < 500 there exists a completely multiplicative sign function with a sum less than that of the Liouville function. Conjecture: A360659(n) < a(n) for n > 20. - Bartlomiej Pawlik, Mar 05 2023
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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.
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).
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LINKS
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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. [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]
Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.
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FORMULA
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MAPLE
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MATHEMATICA
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Accumulate[Join[{0}, LiouvilleLambda[Range[90]]]] (* Harvey P. Dale, Nov 08 2011 *)
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PROG
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(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
(Python)
from functools import reduce
from operator import ixor
from sympy import factorint
def A002819(n): return sum(-1 if reduce(ixor, factorint(i).values(), 0)&1 else 1 for i in range(1, n+1)) # Chai Wah Wu, Dec 19 2022
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CROSSREFS
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KEYWORD
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nice,sign
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AUTHOR
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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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STATUS
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approved
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