%I #14 Feb 09 2023 01:55:22
%S 1,0,0,-1,0,0,0,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,1,
%T 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,
%U 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1
%N Dirichlet inverse of A355448.
%C Multiplicative because A355448 is.
%H Antti Karttunen, <a href="/A355684/b355684.txt">Table of n, a(n) for n = 1..100000</a>
%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A355448(n/d) * a(d).
%F Multiplicative with a(p) = 0, and for e > 1, a(p^e) = -1 if e == 2 or 3 (mod 4) and 1 otherwise. - _Amiram Eldar_, Feb 09 2023
%t f[p_, e_] := If[Mod[e, 4] < 2, 1, -1]; f[p_, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 09 2023 *)
%o (PARI)
%o A355448(n) = (1==gcd(numdiv(n^2),6));
%o memoA355684 = Map();
%o A355684(n) = if(1==n,1,my(v); if(mapisdefined(memoA355684,n,&v), v, v = -sumdiv(n,d,if(d<n,A355448(n/d)*A355684(d),0)); mapput(memoA355684,n,v); (v)));
%Y Cf. A112526 (apparently gives the absolute values), A355448.
%K sign,mult
%O 1
%A _Antti Karttunen_, Jul 14 2022
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