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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

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.

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[3]])/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[1]

        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.)