A035143 - OEIS

%I #11 Nov 18 2023 06:31:02

%S 1,2,2,3,0,4,2,4,3,0,0,6,0,4,0,5,2,6,0,0,4,0,0,8,1,0,4,6,0,0,0,6,0,4,

%T 0,9,2,0,0,0,0,8,0,0,0,0,1,10,3,2,4,0,2,8,0,8,0,0,2,0,2,0,6,7,0,0,0,6,

%U 0,0,2,12,0,4,2,0,0,0,2,0,5

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -47.

%H G. C. Greubel, <a href="/A035143/b035143.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 18 2023: (Start)

%F a(n) = Sum_{d|n} Kronecker(-47, d).

%F Multiplicative with a(47^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-47, p) = -1, and a(p^e) = e+1 if Kronecker(-47, p) = 1.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5*Pi/sqrt(47) = 2.291241... . (End)

%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-47, #] &]];

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

%o (PARI) my(m=-47); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(-47, d)); \\ _Amiram Eldar_, Nov 18 2023

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_