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 A301429 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers. 2
 6, 3, 8, 9, 0, 9, 4, 0, 5, 4, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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)). LINKS 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 FORMULA 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 EXAMPLE 0.638909... MAPLE 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; MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A003136, A003627, A064533, A301430. Sequence in context: A199447 A273067 A306774 * A198836 A271179 A220085 Adjacent sequences:  A301426 A301427 A301428 * A301430 A301431 A301432 KEYWORD nonn,more,cons,hard AUTHOR Michel Waldschmidt, Mar 21 2018 EXTENSIONS Offset corrected by Vaclav Kotesovec, Mar 25 2018 a(6)-a(10) from Peter Luschny, Mar 29 2018 STATUS approved

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Last modified October 14 02:19 EDT 2019. Contains 327994 sequences. (Running on oeis4.)