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A333240
Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2).
18
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
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
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.
Jerzy Kaczorowski, Waldemar Ratajczak, Peter Nijkamp, Krzysztof Górnisiewicz, Economic hierarchical spatial systems - new properties of Löschian numbers, Applied Mathematics and Computation, Volume 461 (2024) 128319. (The product appears in Theorem 3 (12)).
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
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