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%I #11 Sep 16 2023 21:03:13
%S 1,1,0,0,1,0,-1,0,0,1,1,0,-1,-1,0,0,1,0,-1,0,0,1,1,0,0,-1,0,0,1,0,-1,
%T 0,0,1,-1,0,-1,-1,0,0,1,0,-1,0,0,1,1,0,0,0,0,0,1,0,1,0,0,1,1,0,-1,-1,
%U 0,0,-1,0,-1,0,0,-1,1,0,-1,-1,0,0,-1,0,-1,0,0,1,1,0,1,-1,0,0,1,0,1,0,0,1,-1,0,-1,0,0,0,1,0,-1,0,0
%N Dirichlet inverse of A102283.
%C Multiplicative because A102283 is.
%H Antti Karttunen, <a href="/A365428/b365428.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} A102283(n/d) * a(d).
%F Multiplicative with a(p) = -Legendre(-3, p), and a(p^e) = 0 for e >= 2. - _Amiram Eldar_, Sep 16 2023
%t f[p_, e_] := If[e == 1, -JacobiSymbol[-3, p], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 16 2023 *)
%o (PARI) A365428(n) = { my(f=factor(n)); prod(k=1,#f~,if(1<f[k,2],0,-kronecker(-3,f[k,1]))); }; \\ (After _Amiram Eldar_'s multiplicative formula).
%o (PARI)
%o A102283(n) = ([0, 1, -1][n%3 + 1]);
%o memoA365428 = Map();
%o A365428(n) = if(1==n,1,my(v); if(mapisdefined(memoA365428,n,&v), v, v = -sumdiv(n,d,if(d<n,A102283(n/d)*A365428(d),0)); mapput(memoA365428,n,v); (v)));
%Y Cf. A102283, A134323, A156277, A359377 (absolute values).
%K sign,mult
%O 1
%A _Antti Karttunen_, Sep 16 2023
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