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A091396 a(n) = Product_{ p | n } (1 + Legendre(2,p) ).

%I A091396 #18 Oct 17 2022 07:05:54
%S A091396 1,1,0,1,0,0,2,1,0,0,0,0,0,2,0,1,2,0,0,0,0,0,2,0,0,0,0,2,0,0,2,1,0,2,
%T A091396 0,0,0,0,0,0,2,0,0,0,0,2,2,0,2,0,0,0,0,0,0,2,0,0,0,0,0,2,0,1,0,0,0,2,
%U A091396 0,0,2,0,2,0,0,0,0,0,2,0,0,2,0,0,0,0,0,0,2,0,0,2,0,2,0,0,2,2,0,0,0,0,2,0,0
%N A091396 a(n) = Product_{ p | n } (1 + Legendre(2,p) ).
%H A091396 Andrew Howroyd, <a href="/A091396/b091396.txt">Table of n, a(n) for n = 1..1000</a>
%F A091396 Multiplicative with a(2^e)=1, a(p^e)=0 if p mod 8=3 or p mod 8=5 and a(p^e)=2 if p mod 8=1 or p mod 8=7. - _Vladeta Jovovic_, Mar 02 2004
%F A091396 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*sqrt(2) * log(1+sqrt(2))/Pi^2 = 0.505167... . - _Amiram Eldar_, Oct 17 2022
%p A091396 with(numtheory); L := proc(n,N) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul((1+legendre(N,t1[i][1])),i=1..nops(t1)); end; [seq(L(n,2),n=1..120)];
%t A091396 a[n_] := If[n == 1, 1, Product[1 + KroneckerSymbol[2, p], {p, FactorInteger[n][[All, 1]]}]];
%t A091396 a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 20 2019 *)
%o A091396 (PARI) a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + kronecker(2, f[i]))} \\ _Andrew Howroyd_, Jul 23 2018
%Y A091396 Cf. A091337, A091379, A091397, A091398, A091399.
%K A091396 nonn,mult
%O A091396 1,7
%A A091396 _N. J. A. Sloane_, Mar 02 2004