A002819 -id:A002819 - cp's OEIS frontend

A055037 Number of numbers <= n with an even number of prime factors (counted with multiplicity).

1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 8, 8, 8, 8, 8, 9, 10, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 25, 26, 27, 28, 28, 29, 29, 30, 30, 31, 32, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 34

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 01 2000

nonn

Partial sums of A065043.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 92.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Polya Conjecture

Cf. A001222, A002819, A008836, A055038, A065043.

Table[Length[Select[Range[n], EvenQ[PrimeOmega[#]] &]], {n, 75}] (* Alonso del Arte, May 28 2012 *)
Accumulate[Table[(LiouvilleLambda[n] + 1)/2, {n, 1, 100}]] (* Vaclav Kotesovec, Aug 18 2025 *)

first(n)=my(s); vector(n,k,s+=1-bigomega(k)%2) \\ Charles R Greathouse IV, Sep 02 2015

from functools import reduce
from operator import ixor
from sympy import factorint
def A055037(n): return sum(1 for i in range(1,n+1) if not (reduce(ixor, factorint(i).values(),0)&1)) # Chai Wah Wu, Jan 01 2023

a(n) = (1/2)*Sum_{k=1..n} (1+lambda(k)) = (1/2)*(n+L(n)), where lambda(n)=A008836(n) and L(n)=A002819(n).

Formula and more terms from Vladeta Jovovic, Dec 03 2001

Offset corrected by Ray Chandler, May 30 2012

A072203 (Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n).

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 5, 6, 7, 6, 7, 6, 7, 8, 7, 6, 5, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 7, 8, 7, 8, 9, 8, 9, 10, 11, 10, 9, 10, 9, 8, 7, 6, 5, 6, 5, 4, 5, 4, 3, 2, 1, 2, 3, 4, 3, 4, 5, 6

Offset: 1

Bill Dubuque (wgd(AT)zurich.ai.mit.edu), Jul 03 2002

A number m is oddly or evenly factored depending on whether m has an odd or even number of prime factors, e.g., 12 = 2*2*3 has 3 factors so is oddly factored.

Polya conjectured that a(n) >= 0 for all n, but this was disproved by Haselgrove. Lehman gave the first explicit counterexample, a(906180359) = -1; the first counterexample is at 906150257 (Tanaka).

G. Polya, Mathematics and Plausible Reasoning, S.8.16.

T. D. Noe, Table of n, a(n) for n = 1..10000
C. B. Haselgrove, A disproof of a conjecture of Polya, Mathematika 5 (1958), pp. 141-145.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
Kyle Sturgill-Simon, An interesting opportunity: the Gilbreath conjecture, Honors Thesis, Mathematics Dept., Carroll College, 2012.
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3:1, 187-189, 1980.

Cf. A028488, A002819, A051470, A066829.

a072203 n = a072203_list !! (n-1)
a072203_list = scanl1 (\x y -> x + 2*y - 1) a066829_list
-- Reinhard Zumkeller, Nov 19 2011

f[n_Integer] := Length[Flatten[Table[ #[[1]], {#[[2]]}] & /@ FactorInteger[n]]]; g[n_] := g[n] = g[n - 1] + If[ EvenQ[ f[n]], -1, 1]; g[1] = 0; Table[g[n], {n, 1, 103}]
Join[{0},Accumulate[Rest[Table[If[OddQ[PrimeOmega[n]],1,-1],{n,110}]]]] (* Harvey P. Dale, Mar 10 2013 *)
Table[1 - Sum[(-1)^PrimeOmega[i], {i, 1, n}], {n, 1, 100}] (* Indranil Ghosh, Mar 17 2017 *)

a(n) = 1 - sum(i=1, n, (-1)^bigomega(i));
for(n=1, 100, print1(a(n),", ")) \\ Indranil Ghosh, Mar 17 2017

from functools import reduce
from operator import ixor
from sympy import factorint
def A072203(n): return 1+sum(1 if reduce(ixor, factorint(i).values(),0)&1 else -1 for i in range(1,n+1)) # Chai Wah Wu, Dec 20 2022

a(n) = 1 - A002819(n). - T. D. Noe, Feb 06 2007

Edited and extended by Robert G. Wilson v, Jul 13 2002

Comment corrected by Charles R Greathouse IV, Mar 08 2010

A175201 a(n) is the smallest k such that the n consecutive values lambda(k), lambda(k+1), ..., lambda(k+n-1) = 1, where lambda(m) is the Liouville function A008836(m).

Original entry on oeis.org

1, 9, 14, 33, 54, 140, 140, 213, 213, 1934, 1934, 1934, 35811, 38405, 38405, 200938, 200938, 389409, 1792209, 5606457, 8405437, 8405437, 8405437, 8405437, 68780189, 68780189, 68780189, 68780189, 880346227, 880346227, 880346227, 880346227, 880346227

Offset: 1

Michel Lagneau, Mar 04 2010

nonn

Short history of conjecture L(n) <= 0 for all n >= 2 by Deborah Tepper Haimo, where L(n) is the summatory Liouville function A002819(n). 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.

			a(1) = 1 and L(1) = 1;
a(2) = 9 and L(9) = L(10)= 1;
a(3) = 14 and L(14) = L(15) = L(16) = 1;
a(4) = 33 and L(33) = L(34) = L(35) = L(36) = 1.

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.

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..44 (terms < 10^13, first 37 terms from Donovan Johnson)
Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, Sign changes in sums of the Liouville function, Math. Comp. 77 (2008), 1681-1694.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.

Cf. A172354, A051470, A028488, A002819, A175202, A066793.

with(numtheory):for k from 0 to 30 do : indic:=0:for n from 1 to 1000000000 while (indic=0)do :s:=0:for i from 0 to k do :if (-1)^bigomega(n+i)= 1 then s:=s+1: else fi:od:if s=k+1 and indic=0 then print(n):indic:=1:else fi:od:od:

Table[k=1;While[Sum[LiouvilleLambda[k+i],{i,0,n-1}]!=n,k++];k,{n,1,30}]
With[{c=LiouvilleLambda[Range[841*10^4]]},Table[SequencePosition[c,PadRight[ {},n,1],1][[All,1]],{n,24}]//Flatten] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, Jul 27 2022 *)

lambda(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.

A090410 Values of L(10^n), where L(n) is the summatory function of the Liouville function A008836(n).

Original entry on oeis.org

1, 0, -2, -14, -94, -288, -530, -842, -3884, -25216, -116026, -342224, -522626, -966578, -7424752, -29445104, -97617938, -271676470, -618117940, -810056106, -6260758462, -34541748676

Offset: 0

Eric W. Weisstein, Nov 30 2003

L(n) for n <= 10^13 is always negative from 906488081 to 10^13. It reaches a record negative value of -3458310 at 8196557476890. It reaches a record positive value of 829 at 906316571 (A051470(829)). - Donovan Johnson, Mar 08 2011

Eric Weisstein's World of Mathematics, Liouville Function

Cf. A002819, A008836.

a(n) = sum(i=1, 10^n, (-1)^bigomega(i)); \\ Michel Marcus, Sep 29 2015

a(n) = A002819(10^n). - Ray Chandler, May 30 2012

a(9)-a(13) from Donovan Johnson, Mar 08 2011
a(14)-a(17) from Hiroaki Yamanouchi, Jul 13 2014
a(18) from Henri Lifchitz, Dec 01 2014
a(19) from Hiroaki Yamanouchi, Sep 28 2015
a(20)-a(21) from Henri Lifchitz, Nov 08 2024

A239122 Partial sums of A061019.

Original entry on oeis.org

1, -1, -4, 0, -5, 1, -6, -14, -5, 5, -6, -18, -31, -17, -2, 14, -3, -21, -40, -60, -39, -17, -40, -16, 9, 35, 8, -20, -49, -79, -110, -142, -109, -75, -40, -4, -41, -3, 36, 76, 35, -7, -50, -94, -139, -93, -140, -188, -139, -189, -138, -190, -243, -189, -134

Offset: 1

Reinhard Zumkeller, Mar 10 2014

sign

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002819.

a239122 n = a239122_list !! (n-1)
a239122_list = scanl1 (+) a061019_list

Accumulate[Table[n*(-1)^PrimeOmega[n],{n,70}]] (* Harvey P. Dale, Apr 02 2015 *)

from functools import reduce
from operator import ixor
from sympy import factorint
def A239122(n): return sum(-i if reduce(ixor, factorint(i).values(), 0)&1 else i for i in range(1,n+1)) # Chai Wah Wu, Jan 03 2023

A253174 For a permutation of the positive integers B={b(k)}, denote L_B(n) = Sum_{k=1..n} lambda(b(k)), where lambda(n) is Liouville's function (A008836). In this sequence, a(n) = L_B(n) in case B = A098550.

Original entry on oeis.org

1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 9, 8, 7, 6, 7, 6, 7, 8, 9, 8, 7, 6, 7, 6, 5, 6, 7, 8, 9, 10, 9, 8, 9, 10, 11, 10, 9, 10, 11, 12, 13, 14, 13, 14, 13, 12, 13, 12, 11, 10, 11, 10, 9, 8, 9, 10, 9, 8, 9, 10, 11, 12, 11, 10, 9, 10, 11, 12, 13, 12

Offset: 1

Vladimir Shevelev, Jan 08 2015

sign

Let n = p^a*q^b*...r^c be the prime power factorization of b(n). Then lambda(n) is
(-1)^s, where s is the sum of exponents a + b + ... + c.
In case B=A000027 (the natural numbers), G. Polya (1919) conjectured that L_B(n)<=0, for n>=2. But this was disproved in 1958 by B. Haselgrove, and in 1980 M. Tanaka found the smallest counterexample, 906150257.
However, for this sequence we conjecture that a(n)>=0 for all n other than 3. A reason for our conjecture is the later appearance of primes in A098550 than in A000027. By our conjecture, among the first N terms of A098550, the terms with odd s are never in the majority, if N is other than 3. Peter J. C. Moses verified the conjecture up to 2.5*10^5 and, moreover, in this range a(n)>0 for n>6.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000
M. Tanaka, A Numerical Investigation on Cumulative Sum of the Liouville Function, Tokyo J. Math. 3 (1980), 187-189.
Eric Weisstein's World of Mathematics, Liouville Function

Cf. A008836, A002819, A098550, A189229, A247085.

(* b = A098550 *) b[n_ /; n <= 3] := n; b[n_] := b[n] = For[bb = Table[b[j], {j, 1, n-1}]; k=4, True, k++, If[FreeQ[bb, k] && !CoprimeQ[k, b[n-2]] && CoprimeQ[k, b[n-1]], Return[k]]]; a[1]=1; a[n_] := a[n] = a[n-1] + LiouvilleLambda[b[n]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 13 2015 *)

a(n+1) - a(n) = lambda(A098550(n+1)).

A360659 a(n) is the minimum sum of a completely multiplicative sign sequence of length n.

Original entry on oeis.org

0, 1, 0, -1, 0, -1, 0, -1, -2, -1, 0, -1, -2, -3, -4, -3, -2, -3, -4, -5, -4, -5, -4, -5, -6, -5, -6, -7, -8, -9, -8, -9, -8, -7, -8, -7, -6, -7, -8, -7, -8, -9, -10, -11, -12, -13, -14, -15, -14, -13, -12, -13, -14, -15, -14, -13, -14, -15, -16, -17, -18, -19

Offset: 0

Bartlomiej Pawlik, Feb 15 2023

sign

A completely multiplicative sign sequence is a sequence S of numbers -1 and +1 such that S(a*b) = S(a)*S(b).
Directly from the definition, the first term of every multiplicative sign sequence is +1.
The number of completely multiplicative sign sequences of length n is 2^A000720(n), since every multiplicative sign sequence S is uniquely determined by the values of S on prime indexes.
Liouville's function A002819(n) gives the sum of the multiplicative sign sequence of length n in which every prime is assigned the value of -1. This is, therefore, an upper bound on a(n). The first difference occurs at n = 14.
Conjecture: For every n >= 22 there exists completely multiplicative sign sequence of length n with the minimal sum, which have +1's on initial primes and -1's on the remaining primes, i.e., its subsequence on primes is ++...+--...- for some number of +1's.
First differences are either 1 or -1. This is because optimum sequences for n cannot have a sum less than one less than those for n-1 and similarly optimum sequences for n cannot have a sum greater than one more than those for n-1. A zero difference is not possible because terms alternate between even and odd. - Andrew Howroyd, Feb 16 2023
Liouville's function (A002819) is unbounded below, hence this sequence is also unbounded below and every negative number occurs in the sequence. - Rémy Sigrist, Feb 17 2023
From David A. Corneth, May 28 2024: (Start)
One might ease the search by setting values with prime indices in (n/2, n] to -1 in the multiplicative function.
Furthermore one can use the squarefree part of n to evaluate the value of composites in the sequence. For example the value of index 60 has the same value as the one at index 15. (End)

			There are eight completely multiplicative sign sequences of length 5: +--+-, +--++, +-++-, +-+++, ++-+-, ++-++, ++++- and +++++. The sums of those sequences are, respectively, -1, 1, 1, 3, 1, 3, 3 and 5. Hence, the minimum sum is equal to -1 and so a(5) = -1.

David A. Corneth, Table of n, a(n) for n = 0..181 (terms to n = 99 from Bartlomiej Pawlik)

Cf. A000720, A002819, A007913, A008836.

F(n, b)={vector(n, k, my(f=factor(k)); prod(i=1, #f~, if(bittest(b, primepi(f[i, 1])-1), 1, -1)^f[i, 2]))}
a(n)={my(m=oo); for(b=0, 2^primepi(n)-1, m=min(m, vecsum(F(n,b)))); m} \\ Andrew Howroyd, Feb 16 2023

from itertools import product
from sympy import primerange, primepi, factorint
def A360659(n):
    a = dict(zip(primerange(n+1),range(c:=primepi(n))))
    return (min(sum(sum(e for p,e in factorint(m).items() if b[a[p]])&1^1 for m in range(1,n+1)) for b in product((0,1),repeat=c))<<1)-n # Chai Wah Wu, May 31 2024

a(n) <= A002819(n).
Conjecture: a(n) < A002819(n) for n > 20.
|a(n) - a(n-1)| = 1 for n > 0. - Andrew Howroyd, Feb 16 2023
From David A. Corneth, May 28 2024: (Start)
a(k^2) = a(k^2-1) + 1 for k >= 1.
a(p) = a(p-1) - 1 for prime p. (End)

A069545 Liouville clusters: the number of successive occurrences of signs in Liouville function lambda(k); a(2n-1) is number of successive positive signs, while a(2n) is number of successive negative signs.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 2, 3, 3, 4, 2, 1, 3, 6, 4, 1, 3, 5, 1, 2, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 2, 3, 1, 4, 1, 2, 1, 3, 2, 1, 5, 1, 2, 1, 4, 3, 1, 3, 1, 1, 1, 4, 1, 3, 1, 2, 2, 1, 3, 2, 1, 2, 1, 2, 5, 3, 7, 3, 1, 1, 1, 2, 2, 1, 4, 4, 1, 2, 1, 7, 2

Offset: 1

Paul D. Hanna, Apr 17 2002

Related open questions. What is the limit of ratio: a(n)/n, as n->infinity? What is frequency distribution of integer k in the sequence; a(n)=k for what set of n?

Essentially this sequence is a run-length encoding of A008836. - Alonso del Arte, Feb 29 2012

			a(6) = 2 because the 6th Liouville cluster consists of 2 successive negative signs: lambda(7) = lambda(8) = (-1).
a(7) = 2 because the 7th Liouville cluster consists of 2 successive positive signs: lambda(9) = lambda(10) = 1.

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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.
Eric Weisstein's World of Mathematics, Liouville Function

Cf. A008836, A002819, A001222, A028260, A026424.

import Data.List (group)
a069545 n = a069545_list !! (n-1)
a069545_list = map length $ group a008836_list
-- Reinhard Zumkeller, Mar 10 2014

max = 227; lambdaClLens = {}; Module[{curr = 1, cl = 1, iter = 2}, While[iter < max, If[LiouvilleLambda[iter] == curr, cl++, AppendTo[lambdaClLens, cl]; curr = (-1)curr; cl = 1]; iter++]]; lambdaClLens (* Alonso del Arte, Feb 29 2012 *)
Length/@Split[LiouvilleLambda[Range[300]]] (* Harvey P. Dale, Jul 02 2017 *)

Related to summatory Liouville function (A002819): L(m)=sum_{k=1, n} (-1)^(k-1)*a(k) where m=sum_{k=1, n} a(k).

Corrected a(46) and a(47), and added terms after that. - Alonso del Arte, Feb 29 2012

A175202 a(n) is the smallest k such that the n consecutive values L(k), L(k+1), ..., L(k+n-1) = -1, where L(m) is the Liouville function A008836(m).

Original entry on oeis.org

2, 2, 11, 17, 27, 27, 170, 279, 428, 5879, 5879, 13871, 13871, 13871, 41233, 171707, 1004646, 1004646, 1633357, 5460156, 11902755, 21627159, 21627159, 38821328, 41983357, 179376463, 179376463, 179376463, 179376463, 179376463, 179376463, 179376463

Offset: 1

Michel Lagneau, Mar 04 2010

nonn

L(n) = (-1)^omega(n) where omega(n) is the number of prime factors of n with multiplicity.

			a(1) = 2 and L(2) = -1;
a(2) = 2 and L(2) = L(3)= -1;
a(3) = 11 and L(11) = L(12) = L(13) = -1;
a(4) = 17 and L(17) = L(18) = L(19) = L(20) = -1.

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.

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..43 (terms < 10^13, first 38 terms from Donovan Johnson)
Peter Borwein, Ron Ferguson, and Michael J. Mossinghoff, Sign changes in sums of the Liouville function. Math. Comp. 77 (2008), 1681-1694.
R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.

Cf. A002819, A008836, A028488, A051470, A066794, A172354, A175201.

with(numtheory):for k from 0 to 30 do : indic:=0:for n from 1 to 1000000000 while (indic=0)do :s:=0:for i from 0 to k do :if (-1)^bigomega(n+i)= -1 then s:=s+1: else fi:od:if s=k+1 and indic=0 then print(n):indic:=1:else fi:od:od:

Table[k=1;While[Sum[LiouvilleLambda[k+i],{i,0,n-1}]!=-n,k++];k,{n,1,30}]

a(15) and a(21) corrected by Donovan Johnson, Apr 01 2013

A212496 a(n) = Sum_{k=1..n} (-1)^(k-Omega(k)) with Omega(k) the total number of prime factors of k (counted with multiplicity).

Original entry on oeis.org

-1, -2, -1, 0, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 5, 6, 5, 6, 7, 8, 7, 8, 9, 8, 9, 8, 9, 10, 11, 10, 9, 8, 7, 6, 7, 8, 7, 8, 7, 8, 9, 10

Offset: 1

Zhi-Wei Sun, May 19 2012

On May 16 2012, Zhi-Wei Sun conjectured that a(n) is positive for each n > 4. He has verified this for n up to 10^10, and shown that the conjecture implies the Riemann Hypothesis. Moreover, he guessed that a(n) > sqrt(n) for any n > 324 (and also a(n) < sqrt(n)*log(log(n)) for n > 5892); this implies that the sequence contains all natural numbers.

Sun also conjectured that b(n) = Sum_{k=1..n} (-1)^(k-Omega(k))/k < 0 for all n=1,2,3,..., and verified this for n up to 2*10^9. Moreover, he guessed that b(n) < -1/sqrt(n) for all n > 1, and b(n) > -log(log(n))/sqrt(n) for n > 2008.

			We have a(4)=0 since (-1)^(1-Omega(1)) + (-1)^(2-Omega(2)) + (-1)^(3-Omega(3)) + (-1)^(4-Omega(4)) = -1 - 1 + 1 + 1 = 0.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, On a pair of zeta functions, preprint, arxiv:1204.6689 [math.NT], 2012-2016.
Zhi-Wei Sun, On the parities of Omega(n)-n, a message to Number Theory List, May 18, 2012.
Zhi-Wei Sun, Table of n, a(n) for n = 1..10^7 (rar-compressed)

Cf. A008836, A002819.

ListTools:-PartialSums([seq((-1)^(k-numtheory:-bigomega(k)),k=1..60)]); # Robert Israel, Jan 03 2023

PrimeDivisor[n_]:=Part[Transpose[FactorInteger[n]],1]
Omega[n_]:=If[n==1,0,Sum[IntegerExponent[n,Part[PrimeDivisor[n],i]],{i,1,Length[PrimeDivisor[n]]}]]
s[0]=0
s[n_]:=s[n]=s[n-1]+(-1)^(n-Omega[n])
Do[Print[n," ",s[n]],{n,1,100000}]
Accumulate[Table[(-1)^(n-PrimeOmega[n]),{n,1000}]] (* Harvey P. Dale, Oct 07 2013 *)

a(n)=sum(k=1,n, (-1)^(bigomega(k)+k)) \\ Charles R Greathouse IV, Jul 31 2016

from functools import reduce
from operator import ixor
from sympy import factorint
def A212496(n): return sum(-1 if reduce(ixor, factorint(i).values(),i)&1 else 1 for i in range(1,n+1)) # Chai Wah Wu, Jan 03 2023

cp's OEIS Frontend

A055037 Number of numbers <= n with an even number of prime factors (counted with multiplicity).

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A072203 (Number of oddly factored numbers <= n) - (number of evenly factored numbers <= n).

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A175201 a(n) is the smallest k such that the n consecutive values lambda(k), lambda(k+1), ..., lambda(k+n-1) = 1, where lambda(m) is the Liouville function A008836(m).

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A090410 Values of L(10^n), where L(n) is the summatory function of the Liouville function A008836(n).

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A239122 Partial sums of A061019.

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A253174 For a permutation of the positive integers B={b(k)}, denote L_B(n) = Sum_{k=1..n} lambda(b(k)), where lambda(n) is Liouville's function (A008836). In this sequence, a(n) = L_B(n) in case B = A098550.

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A360659 a(n) is the minimum sum of a completely multiplicative sign sequence of length n.

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