%I #12 Nov 17 2023 07:34:39

%S 1,0,0,1,0,0,2,0,1,0,2,0,1,0,0,1,2,0,2,0,0,0,0,0,1,0,0,2,2,0,2,0,0,0,

%T 0,1,0,0,0,0,0,0,0,2,0,0,2,0,3,0,0,1,2,0,0,0,0,0,2,0,2,0,2,1,0,0,2,2,

%U 0,0,2,0,0,0,0,2,4,0,0,0,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -13.

%H Amiram Eldar, <a href="/A035177/b035177.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 17 2023 (Start)

%F a(n) = Sum_{d|n} Kronecker(-13, d).

%F Multiplicative with a(13^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-13, p) = -1, and a(p^e) = e+1 if Kronecker(-13, p) = 1 (p is in A296926).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(13)) = 0.5808806... . (End)

%t a[n_] := DivisorSum[n, KroneckerSymbol[-13, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 17 2023 *)

%o (PARI) my(m = -13); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(-13, d)); \\ _Amiram Eldar_, Nov 17 2023

%Y Cf. A296926.

%K nonn,easy,mult

%O 1,7

%A _N. J. A. Sloane_.