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A035203
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Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 21.
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23
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1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 2, 0, 0, 2, 1, 0, 0, 0, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 2, 1
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OFFSET
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1,5
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COMMENTS
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Coefficients of Dedekind zeta function for the quadratic number field of discriminant 21. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022
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LINKS
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FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log((5+sqrt(21))/2)/sqrt(21) = 0.683807... . - Amiram Eldar, Oct 11 2022
a(n) = Sum_{d|n} Kronecker(21, d).
Multiplicative with a(p^e) = 1 if Kronecker(21, p) = 0 (p = 3 or 7), a(p^e) = (1+(-1)^e)/2 if Kronecker(21, p) = -1 (p is in A038894), and a(p^e) = e+1 if Kronecker(21, p) = 1 (p is in A038893 \ {3, 7}). (End)
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MATHEMATICA
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a[n_] := DivisorSum[n, KroneckerSymbol[21, #] &]; Array[a, 100] (* Amiram Eldar, Oct 11 2022 *)
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PROG
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(PARI) my(m = 21); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(21, d)); \\ Amiram Eldar, Nov 19 2023
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CROSSREFS
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Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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