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A318983
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a(n) = Sum_{d|n} Kronecker(-163, d).
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5
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1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0
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OFFSET
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1,41
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COMMENTS
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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 = -163.
Half of the number of integer solutions to x^2 + x*y + 41*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-163)] counted up to association.
Inverse Moebius transform of A011615.
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LINKS
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FORMULA
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a(n) is multiplicative with a(163^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-163, p) = -1, a(p^e) = e + 1 if Kronecker(-163, p) = 1.
G.f.: Sum_{k>0} Kronecker(-163, k) * x^k / (1 - x^k).
A318985(n) = 2 * a(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(163) = 0.246068... . - Amiram Eldar, Dec 16 2023
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EXAMPLE
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G.f. = x + x^4 + x^9 + x^16 + x^25 + x^36 + 2*x^41 + 2*x^43 + 2*x^47 + x^49 + 2*x^53 + 2*x^61 + x^64 + 2*x^71 + ...
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MATHEMATICA
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a[n_] := DivisorSum[n, KroneckerSymbol[-163, #] &]; Array[a, 100] (* Amiram Eldar, Dec 16 2023 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, kronecker(-163, d))
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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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