%I #22 Jan 22 2021 02:14:29
%S 1,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,
%T 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0
%N a(n) is multiplicative with a(p^e) = -1 if e=2, a(p^e) = 0 if e=1 or e>2.
%H Antti Karttunen, <a href="/A271102/b271102.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F Dirichlet g.f.: 1 / zeta(2*s).
%F abs(a(n)) = A227291(n).
%F Moebius transform of A008966. A008966(n) = abs(mu(n)) = Sum_{d|n} a(d), where mu is the Moebius function (A008683). - _Michael Somos_, Jan 30 2020
%F a(n) = Sum_{d|n} mu(d)*mu(n/d)^2. - _Ridouane Oudra_, Jan 21 2021
%e G.f. = x - x^4 - x^9 - x^25 + x^36 - x^49 + x^100 - x^121 - x^169 + ...
%t Table[Times @@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> If[e == 2, -1, 0]], {n, 105}] (* _Michael De Vlieger_, Jul 29 2017 *)
%t Table[DivisorSum[n, Abs[MoebiusMu[#]]*MoebiusMu[n/#] &], {n, 1, 100}] (* _Vaclav Kotesovec_, Apr 08 2019 *)
%o (PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 - X^2 )[n])};
%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; -(e==2)))};
%o (Scheme) (define (A271102 n) (if (= 1 n) n (* (if (= 2 (A067029 n)) -1 0) (A271102 (A028234 n))))) ;; _Antti Karttunen_, Jul 28 2017
%Y Cf. A008966, A227291 (gives the absolute values), Dirichlet inverse of A010052.
%K sign,mult
%O 1
%A _Michael Somos_, Mar 30 2016
%E More terms from _Antti Karttunen_, Jul 28 2017