A035152 - OEIS

%I #11 Nov 17 2023 11:21:46

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

%T 0,3,2,1,4,0,0,4,0,0,0,2,2,2,3,1,4,2,2,4,0,2,2,2,2,0,0,0,6,1,0,0,2,2,

%U 4,0,0,3,2,2,2,1,0,4,0,0,5

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -38.

%H G. C. Greubel, <a href="/A035152/b035152.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(-38, d).

%F Multiplicative with a(p^e) = 1 if Kronecker(-38, p) = 0 (p = 2 or 19), a(p^e) = (1+(-1)^e)/2 if Kronecker(-38, p) = -1 (p is in A191069), and a(p^e) = e+1 if Kronecker(-38, p) = 1 (p is in A191028).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*Pi/sqrt(38) = 1.5289008... . (End)

%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-38, #] &]];

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

%o (PARI) my(m=-38); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(-38, d)); \\ _Amiram Eldar_, Nov 17 2023

%Y Cf. A191028, A191069.

%K nonn,easy,mult

%O 1,3

%A _N. J. A. Sloane_