A035227 - OEIS

%I #9 Nov 20 2023 11:52:53

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

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

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

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 45.

%H Amiram Eldar, <a href="/A035227/b035227.txt">Table of n, a(n) for n = 1..10000</a>

%F From _Amiram Eldar_, Nov 20 2023: (Start)

%F a(n) = Sum_{d|n} Kronecker(45, d).

%F Multiplicative with a(p^e) = 1 if Kronecker(45, p) = 0 (p = 3 or 5), a(p^e) = (1+(-1)^e)/2 if Kronecker(45, p) = -1 (p is in A003631 \ {3}), and a(p^e) = e+1 if Kronecker(45, p) = 1 (p is in A045468).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 8*log(phi)/(3*sqrt(5)) = 0.573878587952..., where phi is the golden ratio (A001622) . (End)

%t a[n_] := DivisorSum[n, KroneckerSymbol[45, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 20 2023 *)

%o (PARI) my(m = 45); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(45, d)); \\ _Amiram Eldar_, Nov 20 2023

%Y Cf. A001622, A003631, A045468.

%K nonn,easy,mult

%O 1,11

%A _N. J. A. Sloane_