A035143 - OEIS

%I #15 Dec 11 2025 22:07:28

%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

%Y Moebius transform gives A011592.

%Y Cf. A191033 (primes decomposing in Q(sqrt(-47))), A191072 (primes remaining inert).

%Y Dedekind zeta functions for imaginary quadratic number fields of discriminants D = -3..-47, -67, -163: A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, A035167, A192013, A035159, A035155, A035151, A035180, A035147, this sequence, A318982, A318983.

%Y Dedekind zeta functions for real quadratic number fields of discriminants D = 5..41: A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, A035223.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_