A035229 - OEIS

%I #11 Nov 20 2023 10:50:12

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

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

%U 0,0,0,4,0,4,0,6,0,0,0,0,1

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

%H Amiram Eldar, <a href="/A035229/b035229.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(47, d).

%F Multiplicative with a(47^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(47, p) = -1 (p is in A038928), and a(p^e) = e+1 if Kronecker(47, p) = 1 (p is in A038927 \ {47}).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(48+7*sqrt(47))/sqrt(47) = 1.331525560401... . (End)

%t a[n_] := DivisorSum[n, KroneckerSymbol[47, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 20 2023 *)

%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 20 2023

%Y Cf. A038927, A038928.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_