A035221 - OEIS

%I #9 Nov 20 2023 11:53:26

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

%T 4,3,0,4,1,8,2,4,0,0,2,4,0,5,3,6,0,3,0,2,0,8,2,0,0,6,2,4,2,7,2,0,2,0,

%U 2,8,0,4,0,0,3,6,0,2,0,10,1

%N Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 39.

%H Amiram Eldar, <a href="/A035221/b035221.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(39, d).

%F Multiplicative with a(p^e) = 1 if Kronecker(39, p) = 0 (p = 3 or 13), a(p^e) = (1+(-1)^e)/2 if Kronecker(39, p) = -1 (p is in A038918), and a(p^e) = e+1 if Kronecker(39, p) = 1 (p is in A038917 \ {3, 13}).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*log(4*sqrt(39)+25)/sqrt(39) = 2.505443727103... . (End)

%t a[n_] := DivisorSum[n, KroneckerSymbol[39, #] &]; Array[a, 100] (* _Amiram Eldar_, Nov 20 2023 *)

%o (PARI) my(m = 39); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

%o (PARI) a(n) = sumdiv(n, d, kronecker(39, d)); \\ _Amiram Eldar_, Nov 20 2023

%Y Cf. A038917, A038918.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_