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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

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

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: A268893 A199447 A273067 * 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 January 20 12:32 EST 2019. Contains 319330 sequences. (Running on oeis4.)