A329484 Dirichlet convolution of the Louiville function with itself.

1, -2, -2, 3, -2, 4, -2, -4, 3, 4, -2, -6, -2, 4, 4, 5, -2, -6, -2, -6, 4, 4, -2, 8, 3, 4, -4, -6, -2, -8, -2, -6, 4, 4, 4, 9, -2, 4, 4, 8, -2, -8, -2, -6, -6, 4, -2, -10, 3, -6, 4, -6, -2, 8, 4, 8, 4, 4, -2, 12, -2, 4, -6, 7, 4, -8, -2, -6, 4, -8, -2, -12, -2

Offset

1,2

Comments

Up to sign this sequence partitions the positive integers in the same way as A008836. Additional interesting partitions exist when values of this sequence are taken into account.

Links

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

Eric Weisstein's World of Mathematics, Liouville Function.

Wikipedia, Liouville function.

Formula

a(n) = Sum_{d|n} A008836(d) * A008836(n/d).

a(n) = tau(n) * lambda(n) = A000005(n) * A008836(n). - Enrique Pérez Herrero, Sep 15 2020

Multiplicative with a(p^e) = (e+1)*(-1)^e, p prime. - Enrique Pérez Herrero, Sep 20 2020

Dirichlet g.f.: zeta(2*s)^2/zeta(s)^2. - Amiram Eldar, Dec 05 2022

Mathematica

a[n_] := DivisorSum[n, LiouvilleLambda[#] * LiouvilleLambda[n/#] &] ; Array[a, 100] (* Amiram Eldar, Jan 18 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, (-1)^bigomega(d) * (-1)^bigomega(n/d))
(PARI) a(n) = {numdiv(n)*(-1)^bigomega(n)} \\ Andrew Howroyd, Sep 15 2020
(Python)
from math import prod
from sympy import factorint
def A329484(n): return prod(-e-1 if e&1 else e+1 for e in factorint(n).values()) # Chai Wah Wu, Dec 23 2022

Crossrefs

Cf. A000005, A008836.

Sequence in context: A167447 A134687 A184395 * A179941 A179942 A000005

Adjacent sequences: A329481 A329482 A329483 * A329485 A329486 A329487

Keyword

sign,mult

Author

Torlach Rush, Jan 17 2020

Status

approved