cp's OEIS Frontend

A333240 Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2).

%I A333240 #51 Dec 04 2024 09:44:24
%S A333240 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,
%T A333240 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,
%U A333240 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
%N A333240 Decimal expansion of Product_{primes p == 2 (mod 3)} 1/(1 - 1/p^2).
%C A333240 The range of product are the primes of the form 3*k - 1 (A003627).
%C A333240 See a comment of _R. J. Mathar_ in A175646.
%H A333240 Peter Luschny, <a href="/A333240/b333240.txt">Table of n, a(n) for n = 1..1000</a>
%H A333240 Thomas Dence and Carl Pomerance, <a href="http://dx.doi.org/10.1023/A:1009753405498">Euler's Function in Residue Classes</a>, Raman. J., Vol. 2 (1998) pp. 7-20, c_3 in formula (1.8) and (5.6), <a href="https://math.dartmouth.edu/~carlp/PDF/paper116.pdf">alternative link</a>.
%H A333240 S. Ettahri, O. Ramare, L. Surel, <a href="https://arxiv.org/abs/1908.06808">Fast multi-precision computation of some Euler products</a>, arxiv:1908.06808 (2019), Section 9.
%H A333240 Jerzy Kaczorowski, Waldemar Ratajczak, Peter Nijkamp, Krzysztof Górnisiewicz, <a href="https://doi.org/10.1016/j.amc.2023.128319">Economic hierarchical spatial systems - new properties of Löschian numbers</a>, Applied Mathematics and Computation, Volume 461 (2024) 128319. (The product appears in Theorem 3 (12)).
%H A333240 R. J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and prime zeta modulo functions for small moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, Zeta_{3,2}(2) in section 3.2.
%F A333240 A333240 * A175646 = (4*Pi^2)/27 = A214549.
%F A333240 A301429 = sqrt(A333240) / 12^(1/4).
%F A333240 Equals Sum_{k>=1} 1/A004612(k)^2. - _Amiram Eldar_, Sep 27 2020
%e A333240 1.414064390892147637565501819079829379907695069393162175039924962423928106992...
%p A333240 z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
%p A333240 x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:
%p A333240 evalf(mul(x(n), n=1..8), 105); # _Peter Luschny_, Jan 17 2021
%t A333240 digits = 104; precision = digits + 10;
%t A333240 prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
%t A333240 Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
%t A333240 Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
%t A333240 Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
%t A333240 gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
%t A333240 pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;
%t A333240 RealDigits[pgv, 10, digits][[1]]
%t A333240 (* _Jean-François Alcover_, Jan 12 2021, after PARI code due to _Artur Jasinski_ *)
%t A333240 z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];
%t A333240 x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;
%t A333240 N[Product[x[n], {n, 8}], 105] (* _Peter Luschny_, Jan 17 2021 *)
%Y A333240 Cf. A003627, A004612, A175646, A214549, A301429, A333239.
%K A333240 nonn,cons,hard
%O A333240 1,2
%A A333240 _Peter Luschny_, May 13 2020
%E A333240 Last 5 digits corrected by _Jean-François Alcover_, Jan 12 2021
%E A333240 Better name by _Peter Luschny_, Jan 17 2021