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 A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares. 2
 3, 0, 2, 3, 1, 6, 1, 4, 2, 3, 5 (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 sums of two squares and are also represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N*(log(N))^(-3/4). 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. FORMULA Equals (3^(1/4)/2^(5/4)) * Pi^(1/2) * (log(2 + sqrt(3)))^(1/4) / Gamma(1/4) * Product_{p == 5, 7, 11 (mod 12), p prime} (1 - 1/p^2)^(-1/2). One can base the definition on p(n) = A167135(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, 3/2, 25/16, 1225/768, 29645/18432, ... -> L. Then A301430 = sqrt(L)*M with M = ((arccosh(2)/6)^(1/4)*Gamma(3/4))/(2*sqrt(Pi)). - Peter Luschny, Mar 29 2018 EXAMPLE 0.302316... MAPLE Digits:= 1000: with(numtheory): B:= evalf(3^(1/4)*Pi^(1/2)*log(2+sqrt(3))^(1/4)/(2^(5/4)*GAMMA(1/4))): for t to 500 do p:=ithprime(t): if `or`(`or`(`mod`(p, 12) = 5, `mod`(p, 12) = 7), `mod`(p, 12) = 11) then B:= evalf(B/(1-1/p^2)^(1/2)) end if end do: B; MATHEMATICA prec := 200; B = N[(Sqrt[Pi] ((3 Log[2 + Sqrt])/2)^(1/4))/(2 Gamma[1/4]), prec]; For[n = 3, n < 50000, n++, p = Prime[n]; If[Mod[p, 12] != 1, B = B / Sqrt[(1 - 1/p) (1 + 1/p)]]] Print[B] (* Peter Luschny, Mar 23 2018 *) PROG (Julia) using Nemo function A301430(bound)     R = RealField(120)     p = sqrt(const_pi(R))     q = R(3/2) * log(R(2) + sqrt(R(3)))     g = R(2) * gamma(R(1/4))     A = (p * ^(q, 1/4)) / g     U = R(1); Q = R(1)     P = R(1); p = ZZ(5)     a = [2, 4, 6]     while p < bound         if isprime(p)             Q *= p             P *= (p*p - U)         end         p += a         a = circshift(a, -1)     end     (Q / sqrt(P)) * A end A301430(ZZ(100000000000)) |> println # Peter Luschny, Mar 29 2018 CROSSREFS Cf. A003136, A064533, A167135, A301429. Sequence in context: A119493 A224317 A032531 * A328311 A143394 A112455 Adjacent sequences:  A301427 A301428 A301429 * A301431 A301432 A301433 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 02:29 EDT 2019. Contains 327995 sequences. (Running on oeis4.)