A007421 -id:A007421 - 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)

A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Nov 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Harry J. Smith)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Characteristic function of A028260 (positions of 1's). Cf. also A026424 (positions of 0's) and A320655.
One less than A007421.
Cf. A003961, A008836, A010052, A038548 (inverse Möbius transform), A046523, A055037 (partial sums), A343784, A347102, A353337, A353338, A353555, A353557, A353629, A353669, A358750, A358752, A353374, A358775, A356163, A356170.
Cf. also A066829, A353374.

A065043 := proc(n)
    if type(numtheory[bigomega](n),'even') then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 26 2013

Table[(LiouvilleLambda[n]+1)/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)

{ for (n=1, 1000, a=1 - bigomega(n)%2; write("b065043.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 04 2009

A065043(n) = (1 - (bigomega(n)%2)); \\ Antti Karttunen, Apr 19 2022

from operator import ixor
from functools import reduce
from sympy import factorint
def A065043(n): return (reduce(ixor, factorint(n).values(),0)&1)^1 # Chai Wah Wu, Jan 01 2023

a(n) = 1 - A001222(n) mod 2.
a(n) = A007421(n) - 1.
a(n) = 1 - A066829(n).
a(A028260(k)) = 1 and a(A026424(k)) = 0 for all k.
Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/(2*zeta(s)). - Enrique Pérez Herrero, Jul 06 2012
a(n) = (A008836(n) + 1)/2. - Enrique Pérez Herrero, Jul 07 2012
a(n) = A001222(2n) mod 2. - Wesley Ivan Hurt, Jun 22 2013
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^2)/(1 - x^n). - Ilya Gutkovskiy, Apr 25 2017
From Antti Karttunen, Dec 01 2022: (Start)
For x, y >= 1, a(x*y) = 1 - abs(a(x)-a(y)).
a(n) = a(A046523(n)) = A356163(A003961(n)).
a(n) = A000035(A356163(n)+A347102(n)).
a(n) = A010052(n) + A353669(n).
a(n) = A353555(n) + A353557(n).
a(n) = A358750(n) + A358752(n).
a(n) = A353374(n) + A358775(n).
a(n) >= A356170(n).
(End)

Corrected by Charles R Greathouse IV, Sep 02 2009

A066829 Parity of Omega(n): a(n) = 1 if n is the product of an odd number of primes; 0 if product of even number of primes.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 1

G. L. Honaker, Jr., Jan 17 2002

From Reinhard Zumkeller, Jul 01 2009: (Start)
The first N Terms are constructed by the following sieving process:
for j:=1 until N do a(j):=0,
for i:=1 until N/2 do
for j:=2*i step i until N do a(j):=1-a(i). (End)
Omega is also written in the OEIS as bigomega. See also comments, references and formulas in A008836 (Liouville's lambda), A007421 and A065043, that all contain the same information as this sequence. - Antti Karttunen, Apr 30 2022

			From _Reinhard Zumkeller_, Jul 01 2009: (Start)
Sieve for N = 30, also demonstrating the affinity to the Sieve of Eratosthenes:
[initial] a(j):=0, 1<=j<=30:
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[i=1] a(1)=0 --> a(j):=1, 2<=j<=30:
0 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
[i=2] a(2)=1 --> a(2*j):=0, 2<=j<=[30/2]:
0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0
[i=3] a(3)=1 --> a(3*j):=0, 2<=j<=[30/3]:
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0
[i=4] a(4)=0 --> a(4*j):=1, 2<=j<=[30/4]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0
[i=5] a(5)=1 --> a(5*j):=0, 2<=j<=[30/5]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0
[i=6] a(6)=0 --> a(6*j):=1, 2<=j<=[30/6]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1
[i=7] a(7)=1 --> a(7*j):=0, 2<=j<=[30/7]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1
[i=8] a(8)=1 --> a(8*j):=0, 2<=j<=[30/8]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1
[i=9] a(9)=0 --> a(9*j):=1, 2<=j<=[30/9]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1
[i=10] a(10)=0 --> a(10*j):=1, 2<=j<=[30/10]:
0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 1
and so on: a(22):=0 in [i=11], a(24):=0 in [i=12], a(26):=0 in [i=13], a(28):=1 in [i=14], and a(30):=1 in [i=15]. (End)

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
S. Ramanujan, Irregular numbers, J. Indian Math. Soc., 5 (1913), 105-106; Coll. Papers 20-21 (provides Dirichlet g.f.)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n
Index entries for sequences generated by sieves

Characteristic function of A026424 (positions of 1's). Cf. also A028260 (its complement, positions of 0's).
Cf. A001222 (bigomega), A007421, A008836, A055038 (partial sums), A065043, A069545 (run lengths), A072203, A349905, A353556, A353558, A358751, A358753.
Cf. A000035.

a066829 = (`mod` 2) . a001222 -- Reinhard Zumkeller, Nov 19 2011

A066829 := proc(n)
    modp(numtheory[bigomega](n) ,2) ;
end proc:
seq(A066829(n),n=1..80) ; # R. J. Mathar, Jul 15 2017

Table[(1-LiouvilleLambda[n])/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)
Table[If[OddQ[PrimeOmega[n]],1,0],{n,120}] (* Harvey P. Dale, Mar 12 2016 *)

A066829(n) = (bigomega(n)%2); \\ Simplified by Antti Karttunen, Apr 30 2022

from sympy import primeomega as Omega
def a(n): return Omega(n)%2
print([a(n) for n in range(1, 105)]) # Michael S. Branicky, Apr 30 2022

from operator import ixor
from functools import reduce
from sympy import factorint
def A066829(n): return reduce(ixor, factorint(n).values(),0)&1 # Chai Wah Wu, Jan 01 2023

a(A026424(n)) = 1; a(A028260(n)) = 0.
Dirichlet g.f.: (zeta(s)^2 - zeta(2*s)) / (2*zeta(s)). [Typo corrected by Vaclav Kotesovec, Jan 30 2024]
a(n) = (1-A008836(n)) / 2. - Corrected by Antti Karttunen, Apr 30 2022
a(m*n) = a(m) XOR a(n). - Reinhard Zumkeller, Aug 28 2008
a(n) = A001222(n) mod 2. - Reinhard Zumkeller, Nov 19 2011
From Antti Karttunen, May 01 & Nov 30 2022: (Start)
a(n) = 1 - A065043(n) = A349905(n) mod 2.
a(n) = A353556(n) + A353558(n).
a(n) = A358751(n) + A358753(n). (End)
a(n) = A000035(A001222(n)). - Omar E. Pol, Apr 09 2025

Corrected and comment added by Reinhard Zumkeller, Jun 26 2009

A174351 a(n) = lambda(Fibonacci(n)).

Original entry on oeis.org

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

Michel Lagneau, Mar 16 2010

			L(Fibonacci(1))= L(Fibonacci(2))= L(1)= 1.
L(Fibonacci(3))= L(2) = -1.
L(Fibonacci(12))= L(144)= 1.

Cf. A000045 A007421, A002819.

A174351 := proc(n)
        A008836(combinat[fibonacci](n)) ;
end proc: # R. J. Mathar, Jul 08 2012

a(n)=(-1)^bigomega(fibonacci(n)) \\ Charles R Greathouse IV, Jun 13 2013

a(n) = A008836(A000045(n)).

Examples edited by Harvey P. Dale, Dec 02 2022

A288118 Lexicographically earliest sequence of distinct positive terms such that, for any i and j > 0, a(ij) != a(i) a(j).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Jun 05 2017

nonn

If we drop the unicity constraint, then we obtain the Liouville's function (A007421).
This sequence is a permutation of the natural numbers (we can always choose the least value not yet seen at prime positions).
Conjecturally:
- the sequence is self-inverse,
- | a(n) - n | <= 1 for any n > 0,
- | a(i)*a(j) - i*j | <> 1 for any i > 0 and j > 0,
- a(n) = n+1 iff a(n+1) = n.
a(6) = a(1) * a(2) * a(3).
This sequence has connections with A288119; here we avoid a(i)*a(j) = a(i*j), there a(i)+a(j) = a(i*j).

			a(1) cannot equal 1 as a(1*1) != a(1)*a(1); a(1) = 2 is acceptable.
a(2) cannot equal a(1); a(2) = 1 is acceptable.
a(3) cannot equal a(1), a(2); a(3) = 3 is acceptable.
a(4) cannot equal a(1)...a(3), a(2)^2; a(4) = 5 is acceptable.
a(5) cannot equal a(1)...a(4); a(5) = 4 is acceptable.
a(6) cannot equal a(1)...a(5), a(2)*a(3); a(6) = 6 is acceptable.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A007421, A288119.

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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A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

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A066829 Parity of Omega(n): a(n) = 1 if n is the product of an odd number of primes; 0 if product of even number of primes.

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A174351 a(n) = lambda(Fibonacci(n)).

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A288118 Lexicographically earliest sequence of distinct positive terms such that, for any i and j > 0, a(i*j) != a(i) * a(j).

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A175702 Convolution square of the Liouville sequence A008836.

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A288118 Lexicographically earliest sequence of distinct positive terms such that, for any i and j > 0, a(ij) != a(i) a(j).