A002053 -id:A002053 - cp's OEIS frontend

A008836 Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1

Offset: 1

N. J. A. Sloane

Coons and Borwein: "We give a new proof of Fatou's theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function. This result is applied to show that for any non-trivial completely multiplicative function from N to {-1,1}, the series sum_{n=1..infinity} f(n)z^n is transcendental over {Z}[z]; in particular, sum_{n=1..infinity} lambda(n)z^n is transcendental, where lambda is Liouville's function. The transcendence of sum_{n=1..infinity} mu(n)z^n is also proved." - Jonathan Vos Post, Jun 11 2008
Coons proves that a(n) is not k-automatic for any k > 2. - Jonathan Vos Post, Oct 22 2008
The Riemann hypothesis is equivalent to the statement that for every fixed epsilon > 0, lim_{n -> infinity} (a(1) + a(2) + ... + a(n))/n^(1/2 + epsilon) = 0 (Borwein et al., theorem 1.2). - Arkadiusz Wesolowski, Oct 08 2013

			a(4) = 1 because since bigomega(4) = 2 (the prime divisor 2 is counted twice), then (-1)^2 = 1.
a(5) = -1 because 5 is prime and therefore bigomega(5) = 1 and (-1)^1 = -1.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
P. Borwein, S. Choi, B. Rooney and A. Weirathmueller, The Riemann Hypothesis: A Resource for the Aficionado and Virtuoso Alike, Springer, Berlin, 2008, pp. 1-11.
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.
P. Ribenboim, Algebraic Numbers, p. 44.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 279.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 3.3.5 on page 99.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 112.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Borwein, R. Ferguson, and M. J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694.
Benoit Cloitre, A tauberian approach to RH, arXiv:1107.0812 [math.NT], 2011.
Michael Coons and Peter Borwein, Transcendence of Power Series for Some Number Theoretic Functions, arXiv:0806.1563 [math.NT], 2008.
Michael Coons, (Non)Automaticity of number theoretic functions, arXiv:0810.3709 [math.NT], 2008.
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]
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
Andrei Vieru, Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions, arXiv:1306.0496 [math.GM], 2015.
H. Walum, A recurrent pattern in the list of quadratic residues mod a prime and in the values of the Liouville lambda function, J. Numb. Theory 12 (1) (1980) 53-56.
Eric Weisstein's World of Mathematics, Liouville Function
Wikipedia, Liouville function
Index to divisibility sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001222, A002053, A007421, A002819 (partial sums), A008683, A010052, A026424, A028260, A028488, A056912, A056913, A065043, A066829, A106400, A156552, A349905, A063647, A101455, A048691.
Möbius transform of A010052.
Cf. A182448 (Dgf at s=2), A347328 (Dgf at s=3), A347329 (Dgf at s=4).

a008836 = (1 -) . (* 2) . a066829  -- Reinhard Zumkeller, Nov 19 2011

A008836 := n -> (-1)^numtheory[bigomega](n); # Peter Luschny, Sep 15 2011
with(numtheory): A008836 := proc(n) local i,it,s; it := ifactors(n): s := (-1)^add(it[2][i][2], i=1..nops(it[2])): RETURN(s) end:

Table[LiouvilleLambda[n], {n, 100}] (* Enrique Pérez Herrero, Dec 28 2009 *)
Table[If[OddQ[PrimeOmega[n]],-1,1],{n,110}] (* Harvey P. Dale, Sep 10 2014 *)

{a(n) = if( n<1, 0, n=factor(n); (-1)^sum(i=1, matsize(n)[1], n[i,2]))}; /* Michael Somos, Jan 01 2006 */

a(n)=(-1)^bigomega(n) \\ Charles R Greathouse IV, Jan 09 2013

from sympy import factorint
def A008836(n): return -1 if sum(factorint(n).values()) % 2 else 1 # Chai Wah Wu, May 24 2022

Dirichlet g.f.: zeta(2s)/zeta(s); Dirichlet inverse of A008966.
Sum_{ d divides n } lambda(d) = 1 if n is a square, otherwise 0.
Completely multiplicative with a(p) = -1, p prime.
a(n) = (-1)^A001222(n) = (-1)^bigomega(n). - Jonathan Vos Post, Apr 16 2006
a(n) = A033999(A001222(n)). - Jaroslav Krizek, Sep 28 2009
Sum_{d|n} a(d) *(A000005(d))^2 = a(n) *Sum{d|n} A000005(d). - Vladimir Shevelev, May 22 2010
a(n) = 1 - 2*A066829(n). - Reinhard Zumkeller, Nov 19 2011
a(n) = i^(tau(n^2)-1) where tau(n) is A000005 and i is the imaginary unit. - Anthony Browne, May 11 2016
a(n) = A106400(A156552(n)). - Antti Karttunen, May 30 2017
Recurrence: a(1)=1, n > 1: a(n) = sign(1/2 - Sum_{dMats Granvik, Oct 11 2017
a(n) = Sum_{ d | n } A008683(d)*A010052(n/d). - Jinyuan Wang, Apr 20 2019
a(1) = 1; a(n) = -Sum_{d|n, d < n} mu(n/d)^2 * a(d). - Ilya Gutkovskiy, Mar 10 2021
a(n) = (-1)^A349905(n). - Antti Karttunen, Apr 26 2022
From Ridouane Oudra, Jun 02 2024: (Start)
a(n) = (-1)^A066829(n);
a(n) = (-1)^A063647(n);
a(n) = A101455(A048691(n));
a(n) = sin(tau(n^2)*Pi/2). (End)

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

Original entry on oeis.org

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

A007421 Liouville's function: parity of number of primes dividing n (with multiplicity).

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

N. J. A. Sloane

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 279.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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]
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.

Cf. A008836, A065043.

a007421 = (2 -) . (`mod` 2) . a001222 -- Reinhard Zumkeller, Nov 10 2011

a[1] = 2; a[n_] := ((-1)^Total[FactorInteger[n][[All, 2]]] + 3)/2; (* or, from version 7 on : *) a[n_] := Boole[ EvenQ[ PrimeOmega[n]]] + 1; (* or *) a[n_] := (LiouvilleLambda[n] + 3)/2; a[1] = 2; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Apr 08 2013, updated Jan 27 2015 *)

a(n) = ((-1)^bigomega(n)+3)/2, where bigomega(n) is the number of prime divisors of the integer n counted with multiplicity.
a(n) = A065043(n) + 1.
a(n) = 2 - A001222(n) mod 2. - Reinhard Zumkeller, Nov 10 2011

More terms from Vladeta Jovovic, Dec 01 2001

A346457 a(n) is the smallest number k such that |Sum_{j=1..k} (-1)^omega(j)| = n, where omega(j) is the number of distinct primes dividing j.

Original entry on oeis.org

1, 4, 5, 8, 9, 32, 77, 88, 93, 94, 95, 96, 99, 100, 119, 124, 147, 148, 161, 162, 189, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 226, 329, 330, 333, 334, 335, 394, 395, 416, 417, 424, 425, 428, 489, 514, 515, 544, 545, 546, 549, 554, 579, 584, 723, 724, 725, 800

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

nonn

Robert Israel, Table of n, a(n) for n = 1..3223

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346456.

N:= 10000: # for values <= N
omega:= n -> nops(numtheory:-factorset(n)):
R:= map(n -> (-1)^omega(n),[$1..10000]):
S:= map(abs,ListTools:-PartialSums(R)):
m:= max(S):
V:= Vector(m):
for i from 1 to N do if S[i] > 0 and V[S[i]] = 0 then V[S[i]]:= i fi od:
convert(V,list); # Robert Israel, Oct 30 2023

Table[k=1;While[Abs[Sum[(-1)^PrimeNu@j,{j,k}]]!=n,k++];k,{n,30}] (* Giorgos Kalogeropoulos, Jul 19 2021 *)

a(n) = my(k=1); while (abs(sum(j=1, k, (-1)^omega(j))) != n, k++); k; \\ Michel Marcus, Jul 19 2021

a(n) = min {k : |Sum_{j=1..k} mu(rad(j))| = n}, where mu is the Moebius function and rad is the squarefree kernel.

cp's OEIS Frontend

A008836 Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).

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

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A007421 Liouville's function: parity of number of primes dividing n (with multiplicity).

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A346457 a(n) is the smallest number k such that |Sum_{j=1..k} (-1)^omega(j)| = n, where omega(j) is the number of distinct primes dividing j.

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A346455 a(n) is the smallest number k such that Sum_{j=1..k} (-1)^omega(j) = n, where omega(j) is the number of distinct primes dividing j.

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A346456 a(n) is the smallest number k such that Sum_{j=1..k} (-1)^omega(j) = -n, where omega(j) is the number of distinct primes dividing j.

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