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A301429
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Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.
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2
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6, 3, 8, 9, 0, 9, 4, 0, 5, 4, 4
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OFFSET
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0,1
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COMMENTS
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This is the decimal expansion of the number alpha such that the number of positive integers <= N which are represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N/sqrt(log(N)).
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LINKS
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Table of n, a(n) for n=0..10.
Etienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017 and Acta Arithmetica, online 15 March 2018.
StackExchange, Iterative calculation of a number-theoretical constant, Mar 24 2018
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FORMULA
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2^(-1/2)*3^(-1/4)*Product_{p == 2 mod 3, p prime} (1 - p^(-2))^(-1/2).
One can base the definition on p(n) = A003627(n). Setting r(n) = (Product_{k=1..n} p(k)^2) / (Product_{k=1..n} (p(k)^2 - 1)) the rational sequence r(n) starts 4/3, 25/18, 605/432, 174845/124416, ... -> L. Then A301429 = sqrt(L)/12^(1/4). - Peter Luschny, Mar 29 2018
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EXAMPLE
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0.638909...
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MAPLE
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Digits:= 1000: A:= 2^(-1/2)*3^(-1/4):
for t to 40000 do p:= ithprime(t): if `mod`(p, 3) = 2 then
A:= evalf(A/(1-1/p^2)^(1/2)) end if end do: A;
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MATHEMATICA
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prec := 200; A = N[2^(-1/2)3^(-1/4), prec];
For[n = 1, n < 50000, n++, p = Prime[n];
If[Mod[p, 3] == 2, A = A / Sqrt[(1 - 1/p) (1 + 1/p)]]]
Print[A] (* Peter Luschny, Mar 23 2018 *)
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PROG
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(Julia)
using Nemo
function A301429(bound)
R = RealField(120)
A = ^(R(12), -1/4)
U = R(1); Q = R(2)
P = R(3); p = ZZ(5)
while p < bound
if isprime(p)
Q *= p
P *= (p*p - U)
end
p += 6
end
(Q/sqrt(P))*A
end
A301429(ZZ(100000000000)) |> println # Peter Luschny, Mar 29 2018
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CROSSREFS
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Cf. A003136, A003627, A064533, A301430.
Sequence in context: A268893 A199447 A273067 * A198836 A271179 A220085
Adjacent sequences: A301426 A301427 A301428 * A301430 A301431 A301432
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KEYWORD
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nonn,more,cons,hard
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AUTHOR
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Michel Waldschmidt, Mar 21 2018
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EXTENSIONS
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Offset corrected by Vaclav Kotesovec, Mar 25 2018
a(6)-a(10) from Peter Luschny, Mar 29 2018
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STATUS
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approved
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