A002819 Liouville's function L(n) = partial sums of A008836.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nice
• sign

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

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

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.
• Benoit Cloitre, A tauberian approach to RH, arXiv preprint arXiv:1107.0812 [math.NT], 2011-2017.
• 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.
• Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.
• Ben Sparks, 906,150,257 and the Pólya conjecture (MegaFavNumbers), SparksMath video (2020)
• 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

Crossrefs

Cf. A008836, A002053, A028488, A239122, A360659.

Programs

Haskell

a002819 n = a002819_list !! n
a002819_list = scanl (+) 0 a008836_list
-- Reinhard Zumkeller, Nov 19 2011

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 *)

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

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

Formula

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

a(n) = Sum_{k=1..floor(sqrt(n))} A002321(floor(n / k^2)). - Daniel Suteu, May 30 2025

Extensions

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