A035151 - OEIS

%I #12 Nov 17 2023 11:21:49

%S 1,2,1,3,2,2,0,4,1,4,2,3,1,0,2,5,0,2,0,6,0,4,0,4,3,2,1,0,0,4,0,6,2,0,

%T 0,3,0,0,1,8,2,0,2,6,2,0,2,5,1,6,0,3,0,2,4,0,0,0,2,6,2,0,0,7,2,4,0,0,

%U 0,0,2,4,0,0,3,0,0,2,2,10,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -39.

%H G. C. Greubel, <a href="/A035151/b035151.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(-39, d).

%F Multiplicative with a(p^e) = 1 if Kronecker(-39, p) = 0 (p = 3 or 13), a(p^e) = (1+(-1)^e)/2 if Kronecker(-39, p) = -1 (p is in A191070), and a(p^e) = e+1 if Kronecker(-39, p) = 1 (p is in A191029).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/sqrt(39) = 2.0122297... . (End)

%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-39, #] &]];

%t Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Apr 25 2018 *)

%o (PARI) my(m=-39); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(-39, d)); \\ _Amiram Eldar_, Nov 17 2023

%Y Cf. A191029, A191070.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_