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A355691 Dirichlet inverse of A320111, number of divisors of n that are not of the form 4k+2.

%I A355691 #15 Dec 30 2022 06:30:49
%S A355691 1,-1,-2,-1,-2,2,-2,0,1,2,-2,2,-2,2,4,1,-2,-1,-2,2,4,2,-2,0,1,2,0,2,
%T A355691 -2,-4,-2,1,4,2,4,-1,-2,2,4,0,-2,-4,-2,2,-2,2,-2,-2,1,-1,4,2,-2,0,4,0,
%U A355691 4,2,-2,-4,-2,2,-2,0,4,-4,-2,2,4,-4,-2,0,-2,2,-2,2,4,-4,-2,-2,0,2,-2,-4,4,2,4,0,-2,2,4,2,4,2,4,-2,-2,-1,-2,-1,-2,-4,-2,0,-8
%N A355691 Dirichlet inverse of A320111, number of divisors of n that are not of the form 4k+2.
%C A355691 Multiplicative because A320111 is.
%H A355691 Antti Karttunen, <a href="/A355691/b355691.txt">Table of n, a(n) for n = 1..100000</a>
%F A355691 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A320111(n/d) * a(d).
%F A355691 Multiplicative with a(2^e) = A010892(e+2) and for a prime p > 2, a(p) = -2, a(p^2) = 1 and a(p^e) = 0 when e > 2. - _Sebastian Karlsson_, Oct 21 2022
%F A355691 Dirichlet g.f.: 4^s/(zeta(s)^2*(1 - 2^s + 4^s)). - _Amiram Eldar_, Dec 30 2022
%t A355691 f[2, e_] := Switch[Mod[e, 6], 0, 0, 1, -1, 2, -1, 3, 0, 4, 1, 5, 1]; f[p_, 1] = -2; f[p_, 2] = 1; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2022 *)
%o A355691 (PARI)
%o A355691 A320111(n) = sumdiv(n,d,(2!=(d%4)));
%o A355691 memoA355691 = Map();
%o A355691 A355691(n) = if(1==n,1,my(v); if(mapisdefined(memoA355691,n,&v), v, v = -sumdiv(n,d,if(d<n,A320111(n/d)*A355691(d),0)); mapput(memoA355691,n,v); (v)));
%Y A355691 Cf. A320111.
%Y A355691 Cf. also A355690.
%Y A355691 Cf. A010892.
%K A355691 sign,mult
%O A355691 1,3
%A A355691 _Antti Karttunen_, Jul 15 2022