A038917 -id:A038917

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A035221

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

+10
1

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

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OFFSET

1,2

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(39, d).

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}).

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

MATHEMATICA

a[n_] := DivisorSum[n, KroneckerSymbol[39, #] &]; Array[a, 100] (* Amiram Eldar, Nov 20 2023 *)

PROG

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

(PARI) a(n) = sumdiv(n, d, kronecker(39, d)); \\ Amiram Eldar, Nov 20 2023

CROSSREFS

Cf. A038917, A038918.

KEYWORD

nonn,easy,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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