A035169 - OEIS

%I #17 Nov 17 2023 17:51:15

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

%T 2,1,2,0,0,0,2,0,0,2,2,0,0,1,1,0,2,0,0,0,4,0,2,0,0,2,0,0,1,1,0,0,0,2,

%U 2,0,2,0,0,0,3,2,2,0,0,2,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -21.

%H G. C. Greubel, <a href="/A035169/b035169.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(-21, d).

%F Multiplicative with a(p^e) = 1 if Kronecker(-17, p) = 0 (p = 3 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(-17, p) = -1, and a(p^e) = e+1 if Kronecker(-17, p) = 1.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(21)) = 0.914068... . (End)

%t a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-21, #] &]]; Table[a[n], {n, 1, 100}] (* _G. C. Greubel_, Jul 17 2018 *)

%o (PARI) my(m = -21); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(-21, d)); \\ _Amiram Eldar_, Nov 17 2023

%K nonn,easy,mult

%O 1,5

%A _N. J. A. Sloane_