A002819 -id:A002819 - cp's OEIS frontend

A327666 a(n) = Sum_{k = 1..n} (-1)^(Omega(k) - omega(k)), where Omega(k) counts prime factors of k with multiplicity and omega(k) counts distinct prime factors.

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

Offset: 1

Ilya Gutkovskiy, Sep 21 2019

nonn

Partial sums of A162511.

			Omega(1) = omega(1) = 0. The difference is 0, so (-1)^0 = 1, so a(1) = 1.
Omega(2) = omega(2) = 1. The difference is 0, so (-1)^0 = 1, which is added to a(1) to give a(2) = 2.
Omega(3) = omega(3) = 1. The difference is 0, so (-1)^0 = 1, which is added to a(2) to give a(3) = 3.
Omega(4) = 2 but omega(4) = 1. The difference is 1, so (-1)^1 = -1, which is added to a(3) to give a(4) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A002321, A002819, A008836, A046660, A069812, A076479, A162511, A174863, A307868.

Table[Sum[(-1)^(PrimeOmega[k] - PrimeNu[k]), {k, n}], {n, 70}]
f[p_, e_] := (-1)^(e - 1); Accumulate @ Table[Times @@ f @@@ FactorInteger[n], {n, 1, 100}] (* Amiram Eldar, Sep 18 2022 *)

seq(n)={my(v=vector(n)); v[1]=1; for(k=2, n, v[k] = v[k-1] + (-1)^(bigomega(k)-omega(k))); v} \\ Andrew Howroyd, Sep 23 2019

from functools import reduce
from sympy import factorint
def A327666(n): return sum(-1 if reduce(lambda a,b:~(a^b), factorint(i).values(),0)&1 else 1 for i in range(1,n+1)) # Chai Wah Wu, Jan 01 2023

a(1) = 1, a(n) = a(n - 1) + (-1)^(Omega(n) - omega(n)) for n > 1.

a(n) ~ c * n, where c = A307868. - Amiram Eldar, Sep 18 2022

A172357 n such that the Liouville function lambda(n) take successively, from n, the values 1,-1,1,-1,1,-1.

Original entry on oeis.org

58, 185, 194, 274, 287, 342, 344, 382, 493, 566, 667, 856, 858, 926, 1012, 1014, 1157, 1165, 1230, 1232, 1234, 1267, 1318, 1385, 1393, 1418, 1482, 1484, 1679, 1681, 1795, 1841, 1915, 1917, 2060, 2062, 2064, 2232, 2340, 2342, 2567, 2569, 2627, 2805, 3013

Offset: 1

Michel Lagneau, Feb 01 2010

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A051470, A028488, A002819.

with(numtheory): for n from 1 to 4300 do;if (-1)^bigomega(n)=1 and (-1)^bigomega(n+1) = -1 and (-1)^bigomega(n+2) = 1 and (-1)^bigomega(n+3) = -1 and (-1)^bigomega(n+4) = 1 and (-1)^bigomega(n+5) = -1 then print(n); else fi ; od;

Transpose[Transpose[#][[1]]&/@Select[Partition[Table[{n, LiouvilleLambda[ n]},{n,3100}],6,1],Transpose[#][[2]]=={1,-1,1,-1,1,-1}&]][[1]] (* Harvey P. Dale, May 19 2012 *)

lambda(n)=(-1)^bigomega(n);
for(n=1,1e4,if(lambda(n)==1&lambda(n+1)==-1&lambda(n+2)==1&&lambda(n+3)==-1&lambda(n+4)==1&&lambda(n+5)==-1,print1(n", "))) /* Charles R Greathouse IV, Jun 13 2011 */

A174050 Primes of the form x^2 + y^2 such that L(x)* L(y) = 1, where L is the Liouville lambda-function A008836.

Original entry on oeis.org

2, 13, 17, 29, 37, 53, 73, 89, 97, 101, 113, 173, 181, 193, 197, 233, 241, 257, 277, 293, 313, 337, 349, 353, 373, 409, 421, 433, 449, 457, 521, 541, 569, 577, 593, 613, 641, 661, 673, 677, 709, 733, 757, 761, 809, 821, 853, 881, 929, 1021, 1033, 1049, 1069

Offset: 1

Michel Lagneau, Mar 06 2010

nonn

One contribution to the set of solutions is from (x,y) where x and y are both prime, see A045637.

Another set of solutions is contributed if (x,y) are both in A026424.

			2 is in the sequence because 2 = 1 + 1 and L(1)*L(1)= (1) *(1) = 1.
13 is in the sequence because 13 = 2^2 + 3^2 and L(2)*L(3)= (-1)*(-1) = 1.
193 is in the sequence because 193 = 12^2 + 7^2 and L(12)*L(7)= (-1)*(-1) = 1.

Cf. A002819, A002313, A174054.

isA174050 := proc(n)
        local x,y ;
        if not isprime(n) then
                return false;
        end if;
        for x from 1 do
                if x^2 > n then
                        return false;
                end if;
                if issqr(n-x^2) then
                        y := sqrt(n-x^2) ;
                        if A008836(x) * A008836(y) = 1 then
                                return true;
                        end if;
                end if;
        end do:
end proc:
for n from 1 to 1100 do
        if isA174050(n) then
                printf("%d,\n",n) ;
        end if;
end do: # R. J. Mathar, Jul 09 2012

lambdaQ[{x_, y_}] := LiouvilleLambda[x]*LiouvilleLambda[y] == 1; Select[ Prime /@ Range[200], Or @@ lambdaQ /@ PowersRepresentations[#, 2, 2] &] (* Jean-François Alcover, Jul 30 2013 *)

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

A174856 Square array read by antidiagonals up. Redheffer type matrix. T(1,1)=1 and T(n,1) = A049240.

Original entry on oeis.org

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

Offset: 1

Mats Granvik, Mar 31 2010

The first column is equal to 0 when n is a square greater than 1. The rest of the array is equal to A143104. The determinant of this array is A002819.

			The array begins:
  1,1,1,1,1,1,1,1,1,1
  1,1,0,0,0,0,0,0,0,0
  1,0,1,0,0,0,0,0,0,0
  0,1,0,1,0,0,0,0,0,0
  1,0,0,0,1,0,0,0,0,0
  1,1,1,0,0,1,0,0,0,0
  1,0,0,0,0,0,1,0,0,0
  1,1,0,1,0,0,0,1,0,0
  0,0,1,0,0,0,0,0,1,0
  1,1,0,0,1,0,0,0,0,1

Cf. A143104, A002819, A174854, A174852.

t[1, 1] = 1; t[n_, 1] := Boole[!IntegerQ[Sqrt[n]]]; t[n_, k_] := Boole[n == 1 || Mod[n, k] == 0]; Table[t[n - k + 1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 05 2013 *)

A212818 Numbers up to 10^n with an even number of not necessarily distinct prime factors, or positive Liouville function.

Original entry on oeis.org

1, 5, 49, 493, 4953, 49856, 499735, 4999579, 49998058, 499987392, 4999941987, 49999828888, 499999738687, 4999999516711

Offset: 0

Martin Renner, May 28 2012

nonn

			a(1) = 5 since up to 10 there are the five numbers 1, 4, 6, 9, 10 with an even number of prime factors, or positive Liouville function.

Eric Weisstein's World of Mathematics, Liouville Function

Cf. A055037 (goes up to n rather than 10^n), A002819, A008836, A028260, A065043, A090410.

zg:=0: zu:=0: G:=[]: U:=[]: k:=0:
for i from 1 to 10^8 do if numtheory[bigomega](i) mod 2 = 0 then zg:=zg+1: else zu:=zu+1: fi: if i=10^k then G:=[op(G),zg]: U:=[op(U),zu]: k:=k+1: fi: od:
print(G);

Table[Length[Select[Range[10^n], EvenQ[PrimeOmega[#]] &]], {n, 0, 5}] (* Alonso del Arte, May 28 2012 *)
Table[Count[LiouvilleLambda[Range[10^n]], 1], {n, 0, 5}] (* Ray Chandler, May 30 2012 *)

a(n) = A011557(n) - A212819(n).
a(n) = (10^n)/2 + A090410(n)/2. - Donovan Johnson, May 30 2012
a(n) = A055037(10^n). - Ray Chandler, May 30 2012

a(9)-a(13) from Donovan Johnson, May 30 2012

A212819 Numbers up to 10^n with an odd number of prime factors, or negative Liouville function.

Original entry on oeis.org

0, 5, 51, 507, 5047, 50144, 500265, 5000421, 50001942, 500012608, 5000058013, 50000171112, 500000261313, 5000000483289

Offset: 0

Martin Renner, May 28 2012

nonn

			a(1) = 5 since up to 10 there are the five numbers 2, 3, 5, 7, 8 with an odd number of prime factors or negative Liouville function.

Eric Weisstein's World of Mathematics, Liouville Function

Cf. A002819, A008836, A026424, A066829, A090410.

zg:=0: zu:=0: G:=[]: U:=[]: k:=0:
for i from 1 to 10^8 do if numtheory[bigomega](i) mod 2 = 0 then zg:=zg+1: else zu:=zu+1: fi: if i=10^k then G:=[op(G),zg]: U:=[op(U),zu]: k:=k+1: fi: od:
print(U);

Table[Count[LiouvilleLambda[Range[10^n]], -1], {n, 0, 5}] (* Ray Chandler, May 30 2012 *)

a(n) = A011557(n) - A212818(n).
a(n) = (10^n)/2 - A090410(n)/2. - Donovan Johnson, May 30 2012
a(n) = A055038(10^n). - Ray Chandler, May 30 2012

a(9)-a(13) from Donovan Johnson, May 30 2012

A332510 a(n) = Sum_{k=1..n} lambda(floor(n/k)), where lambda = A008836.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 1, 2, 5, 2, 1, 4, 5, 4, 3, 4, 3, 6, 7, 6, 7, 4, 3, 8, 7, 6, 7, 8, 9, 8, 9, 10, 11, 8, 5, 10, 9, 10, 11, 10, 9, 12, 13, 12, 13, 12, 11, 16, 17, 12, 13, 12, 13, 16, 13, 14, 15, 14, 13, 16, 15, 16, 17, 20, 19, 18, 19, 18, 19, 14, 15, 22, 23, 22, 19, 22, 21, 20, 21, 20, 23, 20, 19, 26, 23

Offset: 1

Ilya Gutkovskiy, Feb 14 2020

nonn

Eric Weisstein's World of Mathematics, Liouville Function

Cf. A000196, A002819, A006218, A008683, A008836, A317625, A332509.

Table[Sum[LiouvilleLambda[Floor[n/k]], {k, 1, n}], {n, 1, 85}]
Table[Floor[Sqrt[n]] - Sum[DivisorSum[k, LiouvilleLambda[# - 1] &, # > 1 &], {k, 1, n}], {n, 1, 85}]
nmax = 85; CoefficientList[Series[(1/(1 - x)) ((EllipticTheta[3, 0, x] - 1)/2 - Sum[LiouvilleLambda[k - 1] x^k/(1 - x^k), {k, 2, nmax}]), {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, (-1)^bigomega(n\k)); \\ Michel Marcus, Feb 14 2020

G.f.: (1/(1 - x)) * ((theta_3(x) - 1) / 2 - Sum_{k>=2} lambda(k-1) * x^k / (1 - x^k)).
a(n) = floor(sqrt(n)) - Sum_{k=1..n} Sum_{d|k, d > 1} lambda(d-1).
Sum_{k=1..n} mu(k) * a(floor(n/k)) = lambda(n).

A346202 a(n) = L(n)^2, where L is Liouville's function.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 16, 9, 4, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 9, 4, 1, 0, 1, 4, 9, 16, 25, 16, 25, 36, 25, 36, 25, 36, 49, 36, 25, 16, 9, 4, 9, 4, 9, 4, 9, 4, 1, 4, 9, 16, 9, 16, 25, 36, 49, 36, 49, 64, 49

Offset: 1

Mats Granvik, Jul 10 2021

nonn

The Riemann Hypothesis is equivalent to the statement that, for every fixed eps > 0, lim_{n->oo} (L(n) / n^(eps + 1/2)) = 0.

Peter Borwein, Stephen Choi, Brendan Rooney, and Andrea Weirathmueller, The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, 2007, page 6, Theorem 1.2.

Cf. A002819, A008836, A028488 (positions of zeros).

L:= proc(n) option remember; `if`(n<1, 0,
     (-1)^numtheory[bigomega](n)+L(n-1))
    end:
a:= n-> L(n)^2:
seq(a(n), n=1..77);  # Alois P. Heinz, Jul 28 2021

Table[Sum[LiouvilleLambda[n], {n, 1, nn}]^2, {nn, 1, 77}]

a008836(n) = (-1)^bigomega(n) \\ after Charles R Greathouse IV in A008836
a(n) = sum(i=1, n, sum(j=1, n, a008836(i)*a008836(j))) \\ Felix Fröhlich, Jul 10 2021

from functools import reduce
from operator import ixor
from sympy import factorint
def A346202(n): return sum(-1 if reduce(ixor, factorint(i).values(),0)&1 else 1 for i in range(1,n+1))**2 # Chai Wah Wu, Dec 20 2022

a(n) = Sum_{i=1..n} Sum_{j=1..n} A008836(i)*A008836(j).

a(n) = A002819(n)^2. - Ilya Gutkovskiy, Jul 10 2021

A175702 Convolution square of the Liouville sequence A008836.

Original entry on oeis.org

1, -2, -1, 4, -3, 2, -1, -4, 9, -2, -5, 0, 1, 6, 3, -8, -3, 2, 7, -4, 1, -2, -1, 12, 1, -10, -5, -8, 13, 10, -1, -12, 1, 6, 3, 0, -7, 6, 11, -8, -3, -6, -1, -4, -3, 2, 7, 12, 21, -14, -5, -16, -7, 22, -5, -8, -3, 2, 19, 16, -7, -10, 7, -4, -3, -22, -9, -12, 13, 10, 7, 12, 5, 10, -9

Offset: 1

Michel Lagneau, Aug 10 2010

sign

Cf. A002819, A007421

with(numtheory): T:=array(1..200):for p from 1 to 200 do: T[p] :=(-1)^bigomega(p): od :for n from 1 to 100 do: printf(`%d, `, sum (T[k]*T[n+1-k],k=1..n)):od:

a(n)= Sum_{k=1..n} lambda(k)*lambda(n+1-k).

Definition slightly rephrased, keyword:sign added - R. J. Mathar, Aug 19 2010

cp's OEIS Frontend

A327666 a(n) = Sum_{k = 1..n} (-1)^(Omega(k) - omega(k)), where Omega(k) counts prime factors of k with multiplicity and omega(k) counts distinct prime factors.

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A172357 n such that the Liouville function lambda(n) take successively, from n, the values 1,-1,1,-1,1,-1.

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A174050 Primes of the form x^2 + y^2 such that L(x)* L(y) = 1, where L is the Liouville lambda-function A008836.

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

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A174856 Square array read by antidiagonals up. Redheffer type matrix. T(1,1)=1 and T(n,1) = A049240.

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A212818 Numbers up to 10^n with an even number of not necessarily distinct prime factors, or positive Liouville function.

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Maple

Mathematica

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A212819 Numbers up to 10^n with an odd number of prime factors, or negative Liouville function.

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A332510 a(n) = Sum_{k=1..n} lambda(floor(n/k)), where lambda = A008836.

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