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A333240 Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2). 17
1, 4, 1, 4, 0, 6, 4, 3, 9, 0, 8, 9, 2, 1, 4, 7, 6, 3, 7, 5, 6, 5, 5, 0, 1, 8, 1, 9, 0, 7, 9, 8, 2, 9, 3, 7, 9, 9, 0, 7, 6, 9, 5, 0, 6, 9, 3, 9, 3, 1, 6, 2, 1, 7, 5, 0, 3, 9, 9, 2, 4, 9, 6, 2, 4, 2, 3, 9, 2, 8, 1, 0, 6, 9, 9, 2, 0, 8, 8, 4, 9, 9, 4, 5, 3, 7, 5, 4, 8, 5, 8, 5, 0, 2, 4, 7, 5, 1, 1, 4, 2, 0, 0, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The range of product are the primes of the form 3*k - 1 (A003627).

See a comment of R. J. Mathar in A175646.

LINKS

Peter Luschny, Table of n, a(n) for n = 1..1000

Thomas Dence and Carl Pomerance, Euler's Function in Residue Classes, Raman. J., Vol. 2 (1998) pp. 7-20, c_3 in formula (1.8) and (5.6), alternative link.

R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{3,2}(2) in section 3.2.

FORMULA

A333240 * A175646 = (4*Pi^2)/27 = A214549.

A301429 = sqrt(A333240) / 12^(1/4).

Equals Sum_{k>=1} 1/A004612(k)^2. - Amiram Eldar, Sep 27 2020

EXAMPLE

1.414064390892147637565501819079829379907695069393162175039924962423928106992...

MAPLE

z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):

x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:

evalf(mul(x(n), n=1..8), 105); # Peter Luschny, Jan 17 2021

MATHEMATICA

digits = 104; precision = digits + 10;

prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];

Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;

Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];

Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];

gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];

pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;

RealDigits[pgv, 10, digits][[1]]

(* Jean-François Alcover, Jan 12 2021, after PARI code due to Artur Jasinski *)

z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];

x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;

N[Product[x[n], {n, 8}], 105] (* Peter Luschny, Jan 17 2021 *)

CROSSREFS

Cf. A003627, A004612, A175646, A214549, A301429, A333239.

Sequence in context: A106141 A082999 A308255 * A010641 A165486 A100086

Adjacent sequences:  A333237 A333238 A333239 * A333241 A333242 A333243

KEYWORD

nonn,cons,hard

AUTHOR

Peter Luschny, May 13 2020

EXTENSIONS

Last 5 digits corrected by Jean-François Alcover, Jan 12 2021

Better name by Peter Luschny, Jan 17 2021

STATUS

approved

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Last modified May 22 00:02 EDT 2022. Contains 353931 sequences. (Running on oeis4.)