A035145 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -45.

1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 2, 0, 1, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 0, 2, 0, 1, 0, 2, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1

Offset

1,7

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

From Amiram Eldar, Nov 18 2023: (Start)

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

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, and a(p^e) = e+1 if Kronecker(-45, p) = 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(9*sqrt(5)) = 0.624427... . (End)

Mathematica

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-45, #] &]];
Table[a[n], {n, 1, 100}] (* G. C. Greubel, Apr 25 2018 *)

Prog

(PARI) my(m=-45); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(-45, d)); \\ Amiram Eldar, Nov 18 2023

Crossrefs

Sequence in context: A193261 A283497 A265507 * A244315 A214303 A281274

Adjacent sequences: A035142 A035143 A035144 * A035146 A035147 A035148

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved