%I #53 Dec 24 2022 03:39:36
%S 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,
%T -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,
%U 8,4,8,4,4,-2,12,-2,4,-6,7,4,-8,-2,-6,4,-8,-2,-12,-2
%N Dirichlet convolution of the Louiville function with itself.
%C 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.
%H Amiram Eldar, <a href="/A329484/b329484.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LiouvilleFunction.html">Liouville Function</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Liouville_function">Liouville function</a>.
%F a(n) = Sum_{d|n} A008836(d) * A008836(n/d).
%F a(n) = tau(n) * lambda(n) = A000005(n) * A008836(n). - _Enrique Pérez Herrero_, Sep 15 2020
%F Multiplicative with a(p^e) = (e+1)*(-1)^e, p prime. - _Enrique Pérez Herrero_, Sep 20 2020
%F Dirichlet g.f.: zeta(2*s)^2/zeta(s)^2. - _Amiram Eldar_, Dec 05 2022
%t a[n_] := DivisorSum[n, LiouvilleLambda[#] * LiouvilleLambda[n/#] &] ; Array[a, 100] (* _Amiram Eldar_, Jan 18 2020 *)
%o (PARI) a(n) = sumdiv(n,d,(-1)^bigomega(d) * (-1)^bigomega(n/d))
%o (PARI) a(n) = {numdiv(n)*(-1)^bigomega(n)} \\ _Andrew Howroyd_, Sep 15 2020
%o (Python)
%o from math import prod
%o from sympy import factorint
%o 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
%Y Cf. A000005, A008836.
%K sign,mult
%O 1,2
%A _Torlach Rush_, Jan 17 2020