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A035147
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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 = -43.
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7
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1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1
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OFFSET
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1,11
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COMMENTS
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Half of the number of integer solutions to x^2 + x*y + 11*y^2 = n.
Inverse Moebius transform of A011591. (End)
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LINKS
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FORMULA
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a(n) is multiplicative with a(43^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-43, p) = -1, a(p^e) = e + 1 if Kronecker(-43, p) = 1.
G.f.: Sum_{k>0} Kronecker(-43, k) * x^k / (1 - x^k).
A138811(n) = 2 * a(n) unless n = 0. (End)
a(n) = Sum_{d|n} Kronecker(-43, d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(43) = 0.479088... . (End)
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MATHEMATICA
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a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-43, #] &]];
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PROG
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(PARI) my(m=-43); direuler(p=2, 101, 1/(1-(kronecker(m, p)*(X-X^2))-X))
(PARI) a(n) = sumdiv(n, d, kronecker(-43, d)); \\ Amiram Eldar, Nov 18 2023
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CROSSREFS
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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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