A035227 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, 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, 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, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 2, 1, 1

Offset

1,11

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

From Amiram Eldar, Nov 20 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 (p is in A003631 \ {3}), and a(p^e) = e+1 if Kronecker(45, p) = 1 (p is in A045468).

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)

Mathematica

a[n_] := DivisorSum[n, KroneckerSymbol[45, #] &]; Array[a, 100] (* Amiram Eldar, Nov 20 2023 *)

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 20 2023

Crossrefs

Cf. A001622, A003631, A045468.

Sequence in context: A083025 A046080 A170967 * A355320 A328556 A321888

Adjacent sequences: A035224 A035225 A035226 * A035228 A035229 A035230

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved