A239122 -id:A239122 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

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

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

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

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

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

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

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

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

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

A061019 Negate primes in factorization of n.

Original entry on oeis.org

1, -2, -3, 4, -5, 6, -7, -8, 9, 10, -11, -12, -13, 14, 15, 16, -17, -18, -19, -20, 21, 22, -23, 24, 25, 26, -27, -28, -29, -30, -31, -32, 33, 34, 35, 36, -37, 38, 39, 40, -41, -42, -43, -44, -45, 46, -47, -48, 49, -50, 51, -52, -53, 54, 55, 56, 57, 58, -59, 60, -61, 62, -63, 64, 65, -66, -67, -68, 69, -70

Offset: 1

Marc LeBrun, Apr 13 2001

Inverse Moebius transform of A158523. - Corrected by Antti Karttunen, Nov 26 2024

			a(6) = (-2)(-3) = +6, while a(8) = (-2)^3 = -8.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000027, A001222, A061020, A001615, A158523 (Möbius transform).
Cf. A027746.
Cf. A239122 (partial sums).

a061019 1 = 1
a061019 n = product $ map negate $ a027746_row n
-- Reinhard Zumkeller, Feb 08 2012

Table[n (-1)^PrimeOmega[n],{n,70}] (* Harvey P. Dale, Oct 05 2011 *)

a(n) = if( bitand(bigomega(n),1), - n, n ); /* Joerg Arndt, Sep 19 2012 */

from functools import reduce
from operator import ixor
from sympy import factorint
def A061019(n): return -n if reduce(ixor, factorint(n).values(),0)&1 else n # Chai Wah Wu, Dec 20 2022

a(n) = n*lambda(n), where lambda is Liouville's function: A008836.
a(n) = (-1)^(number of primes dividing n)*n = n * (-1)^A001222(n) = n*A008836(n).
Totally multiplicative with a(p) = -p for prime p. [Jaroslav Krizek, Nov 01 2009]
Dirichlet g.f.: zeta(2*s-2)/zeta(s-1). Dirichlet inverse of A055615, all terms turned positive there. - R. J. Mathar, Apr 16 2011
a(n) = Sum_{d|n} lambda(d)*psi(d) = sum_{d|n} A008836(d)* A001615(d) = n/lambda(n). - Enrique Pérez Herrero, Sep 18 2012

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

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A061019 Negate primes in factorization of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula